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Inverse Problems in Science and Engineering
ISSN: 1741-5977 (Print) 1741-5985 (Online) Journal homepage: http://www.tandfonline.com/loi/gipe20
Multi-level backcalculation algorithm for robust
determination of pavement layers parameters
Tomasz Garbowski & Andrzej Pożarycki
To cite this article: Tomasz Garbowski & Andrzej Pożarycki (2016): Multi-level backcalculation
algorithm for robust determination of pavement layers parameters, Inverse Problems in
Science and Engineering, DOI: 10.1080/17415977.2016.1191073
To link to this article: http://dx.doi.org/10.1080/17415977.2016.1191073
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INVERSE PROBLEMS IN SCIENCE AND ENGINEERING, 2016
http://dx.doi.org/10.1080/17415977.2016.1191073
Multi-level backcalculation algorithm for robust
determination of pavement layers parameters
Tomasz Garbowskiaand Andrzej Po˙zaryckib
aInstitute of Structural Engineering, Poznan University of Technology, Poznan, Poland; bInstitute of Civil
Engineering, Poznan University of Technology, Poznan, Poland
ABSTRACT
The backcalculation procedure applied to a mechanical chara-
cterization of the road pavement is usually limited to an identification
of the elastic modulus of each layer only. The remaining parameters
of the model are usually set as known, while performing an inverse
analysis. Among assumed parameters are thicknesses of the model
layers and frequently considered as constants on a homogeneous
section of the road. This is an obvious simplification, because sections
in general are inhomogeneous, i.e. the thickness of each layer changes
slightly along each road section. Thus the precise and possibly
nondestructive estimation of the layers thicknesses is very important
and crucial in the inverse procedure. Here, a hybrid form of the
optimization algorithm, where the condition of a constant thickness
does not need to be fulfilled is described. Further on, the objective
function is formed as a discrepancy between reference and computed
deflection derivative instead of a deflection curve. In consequence, the
values of backcalculated parameters are several per cent more precise
compared to a standard procedure. Whenever the thickness of the
asphalt layer of the pavement structure cannot be assumed a priori as
a constant, the proposed here method appears to be necessary if one
does not want to perform costly and destructive in situ drilling tests.
ARTICLE HISTORY
Received 15 January 2016
Accessed 14 May 2016
KEYWORDS
Inverse analysis; pavement
design; falling weight
deflectomiter; deflection
basin; deflection curve’s
derivative
1. Introduction
The backcalculation analysis is an essential tool of the mechanistic analysis of the road
pavements.[1,2] In general, the purpose of an inverse procedure is to determine the causal
parameters from the observations of the structure. The measurements in a frame of an
in situ pavement testing are often limited to superficial carriageway deflections, while the
parameters to be determined are usually stiffness and thickness of each layer. The typical
tool applied for the nondestructive evaluation of pavement deflection bowl under the
dynamic load is the falling weight deflectometer (FWD). This device is widely used in a civil
engineering practice.[3–7] The results of the backcalculation are obtained by minimizing
the discrepancy between easily measurable pavement quantities and the corresponding
values calculated using analytical or numerical pavement models, e.g. method of equivalent
thickness (MET),[8] layered elastic theory (LET),[9] finite element method (FEM),[10]
boundary element method (BEM),[11] spectral element method (SEM) [12]).
CONTACT Tomasz Garbowski tomasz.garbowski@put.poznan.pl
© 2016 Informa UK Limited, trading as Taylor & Francis Group
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2 T. GARBOWSKI AND A. PO ˙
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Each homogeneous layer of the road pavement model can be fully described by the
constant thickness and constitutive properties. Due to the relatively simple geometry of
carriageway pavements, the analytical or numerical model are most often described in the
cylindrical coordinate system. This obviously determines the form of the equilibrium equa-
tions of the axisymmetric multilayered half-space. The differential equations describing
the mechanics of such models can be found e.g. in [13,14]. In the simple but widely used in
the engineering content case of the elastic multilayered model under a static load, totally
four independent set of quantities are to be determined, namely: (1) the elastic modulus of
each layer, (2) the Poisson’s ratio of each layer, (3) the thickness of each layer, and (4) the
parameters defining the quality of the bonding between adjacent layers.
Even in such simplified model, the growing number of layers implies ill-conditioning
of the inverse problem from the mathematical perspective. The reason for this issue
lies in a well-known compensation effect,[15] which causes that practically any pair of
parameters (i.e. layer stiffness and its thickness) can produce similar responses in terms
of computed deflections. For this reason, it is common to look for either the values
of the elasticity modulus of the model [14,16] or their thickness.[17] By a decoupling
of the model parameters into known a priori and sought quantities another important
sources of inaccurate identification arise. The most essential ones are due to an imprecise
determination of e.g. thickness of the pavement layers [18,19] or a wrong construction or
regularization of the objective function.
In the literature, one can find also approaches to identify a full set of model parameters,
namely both thicknesses and stiffnesses. The most popular tool used in this content is the
artificial neural network (ANN).[20] In this case, limiting the number of simultaneously
identifiable parameters has less importance, whereas the crucial is a quality of the database
[21] and the performance of the training algorithm. With an appropriately trained ANN
structure, it is possible to determine the elastic moduli, thicknesses, and the Poisson’s
ratios at the same time.[20] According to the conclusions driven by some authors, e.g.
[19], the ANN method is also resistant to the noise resulting from deflection and thickness
measurements in-situ. Slightly different approach based on the data mining techniques,
proposed in the work [22] provide the value of backcalculation error by several times lower,
compared to the values achieved with the use of the linear regression or ANN.
There is also another group of methods, currently under development, in which the
thickness of pavement layers is included in a merely indirect way. These methods determine
the so-called global parameter,[23] e.g. in the form of a structural number SN (introduced
by AASHTO). Some researchers also use deflection basin parameters (DBPs), e.g. surface
curvature index (SCI), base damage index (BDI) and base curvature index (BCI).[24]The
fact, that these parameters are in strong correlation with the equivalent pavement stiffness
Eeq, is the basis for simple and empirical backcalculation formulas. However, these methods
are developed for a pavement diagnosis purpose and can only serve as a supplement to
mechanistic methods.
A disappointing disadvantage of standard procedures in a backcalculation scheme is, the
already mentioned, the compensation effect.[15] The result, typical for this phenomenon,
can be observed during the careful study of the determined parameters of the pavement
model in terms of their values and corresponding discrepancies. Which means that two
substantially different converged solutions can result in fairly similar values of the objective
function. It becomes then evident that the low values of the mean square error do not, in
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INVERSE PROBLEMS IN SCIENCE AND ENGINEERING 3
this case, guarantee reaching the real sought values from an engineering perspective. The
use of non-linear elastic constitutive formulation in the pavement multilayer model is
considered as a solution to this problem.[15] It is also possible to introduce to the model
the so-called stiff layer (bedrock) in order to circumvent the ill-condition formulation of
the inverse problem.
The most important ingredients in the inverse analysis are a well-fitted minimization
algorithm, which should provide robust and rapid solution. An interesting approach for
mechanical identification of the multilayered strata, which is based on a robust system
identification method (SIM) is proposed in [25,26]. In both works, the inverse procedure
for transverselly isotropic elastic material calibration is utilized. A combination of the dy-
namic FEM model and nonlocal optimization techniques based on the genetic algorithms
is presented in [27], which provides a global minimum of the minimization problem.
However, even the most suited algorithms suffer from escalating results of the in-situ
deflection measurements. A partial solution to this problem is described in [4], where
authors used a probabilistic beckcalculation method combined with a simulation of a
measurement test of pavement deflections caused by a dynamic loading. Authors of the
work [18] suggest, moreover, that the form of the objective function has a direct influence
on the accuracy of backcalculation results. A considerable improvement of the results can
be gained by exchanging the root mean square error (RMSE) formula with the area value
with correction factor (AVCF) measure, which slightly regularize the problem.
In the commonly used backcalculation procedures of pavement characterization, it is
assumed that the thickness of each layer is known. It is also assumed that for the subsequent
uniform sections of the pavement they are constant values. In the engineering practice, both
the limited accuracy in paving construction technology and the inaccuracies in pavement
layers’ thicknesses characterization are the well-known problems, described, e.g. in the
work.[28,29] Authors studied the uniform 50-meter long road section and demonstrated
that the difference between outliers of the total thickness of the asphalt layers equalled to
3 cm. Therefore, it is very unlikely to conduct an exact synchronization of the real layer
thickness (measured or interpolated) with the current location of the in situ deflection
curve measurement. Assumption that the thicknesses of the particular model?s layers are
constant along the uniform section gives in practice wrong results, which is demonstrated
in the conclusions of the work.[30] All those facts imply the need to popularize optimization
algorithms in which the thickness of the first layer of the pavement model is determined
simultaneously with the values of elastic modulus of each layer.
This paper consists of two separate lines of reasoning and corresponding imple-
mentations. The first one is to develop the new computer algorithm capable to fulfil
the specified requirements, namely to investigate simultaneously both thicknesses and
stiffnesses of the layered pavement structure. Here, the effectiveness of the multilevel mini-
mization algorithm is checked. The first level (outer loop) is responsible for the convergence
of the asphalt layer’s thickness while the second level (inner loop) is designed to conduct
the traditional investigation of elastic modulus of each layer having thicknesses fixed.
In the second thread, the authors discuss the limitations of the backcalculation procedure,
based exclusively on the shape of deflection curves. A hypothesis is made that due to
the transformation of deflection curves into its derivative curves will improve the accuracy
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4 T. GARBOWSKI AND A. PO ˙
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Table 1. Set of the parameters for two different pavement models analysed in the paper.
Model A Model B
E1[MPa] 7.000 9.368
E2[MPa] 300 300
E3[MPa] 100 100
h1[m] 0.20 0.18
h2[m] 0.30 0.30
of backcalculation results. The form of the objective function is then expressed with an
equation:
RMSE =f∂ui
∂ri.
2. Methodology
2.1. Pseudo-experimental models
Here, the two fairly similar pavement structures are considered (see Table 1). The pave-
ment’s models have a relatively different set of parameters, which are chosen so that they
have possibly identical deflection curves. The assumed values of the Poisson’s ratios are
constant, and are equal to ν= 0.3. The value for all layer’s parameters in the models is
shown in Table 1. It is also assumed that there is a full bonding between the model layers.
All pseudo-experimental data (used here to mimic the real measurements) were gen-
erated by an axisymmetric finite element model with a horizontal and vertical dimension
equal to 15 m and 30 m, respectively (detailed studies on the model size are given in
[31]). Four nodal axisymmetric finite elements with a full integration scheme are used here
along with linear elastic constitutive relationship implemented in each pavement layer.
All pseudo-experimental data are subsequently perturbed by random normal distribution
of a noise within an interval ±1µm. Numerical pseudo-experimental data have, beside
the obvious ease in generating, also another very important feature, i.e. the identified
parameters from inverse analysis can be straightforwardly compared with those used to
generate the model response. The computational tools both for direct and inverse analysis
used in this work are the part of the FEMat scientific toolbox.
2.2. Multi-level backcalculation algorithm
Because the backcalculation of the multilayered pavement structure basing on deflections
measurements is in general ill-posed, the determination of layer moduli Eirequires having
all layers’ thicknesses hifixed to the known values. Despite the fact that such logic can
be justified, it is still very sensitive to small errors in layer’s thickness estimation. For
example, ±5% error in h1estimation generates ±15% error in an backcalculation of the
elastic modulus E1. Same error of h2gives proportional error in E2characterization. The
above observation inspires to design the backcalculation algorithm which will be able to
automatically update the model thicknesses to their correct values if initially assumed
values are wrong. Here, the investigation is limited to the first layer’s thickness correction
only. The general idea of the proposed here procedure is presented in Figure 1.
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INVERSE PROBLEMS IN SCIENCE AND ENGINEERING 5
Figure 1. The scheme used in MLBA procedure.
In order to bring the details of the proposed multi-level backcalculatoin algorithm
(MLBA) one needs to discuss also the general framework of inverse analysis and mini-
mization algorithm. Herein, the brief explanation of main features of the inverse procedure
followed by a detailed elucidation of implemented minimization algorithms is presented.
Backcalculation analysis with a particular application to constitutive model calibration
is a tool widely used by many researchers.[32–39] In general it merges the numerically
computed UNUM and experimentally determined UEXP measurable quantities for a dis-
crepancy minimization. A vector of residua Rcan be constructed in the following way:
R=UEXP −UNUM(x). (1)
This measures the differences between the aforementioned measurable quantities. By
adjusting the constitutive parameters (encapsulated in the vector x) embedded in the
numerical model, which in turn mimic the experimental setup, an iterative convergence
towards the required solution can be achieved. The minimization of the objective function
ω(within the least square frame) takes the form:
ω=
n
i=1Ri2=R2
2,(2)
and is usually updated through the use of first-order (gradient-based) or zero-order
(gradient-less) algorithms. Procedures based on a soft method (e.g. genetic algorithms, sim-
ulated annealing, particle swarm algorithms) can be also used for function minimization,
especially when the function is non-convex and so has many local minima. However, such
algorithms usually require many iterations. Among the many first-order procedures that
are based on either the Gauss-Newton or the steepest descent direction in a nonlinear least
square methods, the trust region algorithm (TRA) seems the most effective. The TRA uses
a simple idea, similar to that in Levenberg-Marquardt (LM) algorithm (see e.g. [40], which
performs each new step in a direction combining the Gauss-Newton and steepest descent
directions. The LM algorithm computes new directions using the following formula:
x=−
Mx+λI−1gx,(3)
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6 T. GARBOWSKI AND A. PO ˙
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where λis an internal parameter, gx=∇ωxis the gradient of the objective function ω
with respect to the parameters x
gx=∂ω
∂x,(4)
and the Hessian Mx=∇
2ωxis a second partial derivative of ωwith respect to the
parameters x:
Mx=∂2ω
∂x2,(5)
In the nonlinear least square approach method, the gradient and Hessian matrix can be
computed using the Jacobian matrix:
Jx=∂R
∂x,(6)
so the gradient and Hessian matrix are defined, respectively:
gx=JTR,MxJTJ.(7)
Such approximation of the Hessian, which can be computed ‘for free’ once the Jacobian
is available, represents a distinctive feature of least squares problems. This approximation
is valid if the residuals are small, meaning we are close to the solution. Therefore some
techniques may be required in order to ensure that the Hessian matrix is semi-positive
defined (see e.g. [40]).
One of the main issues of the trust region approach, which to a large extent determines
the success and the performance of this algorithm, is in deciding how large the trusted
region should be. Allowing it to be too large can cause the algorithm to face the same
problem as the classical Newton direction line search, when the model function minimizer
is quite distinct from the minimizer of the actual objective function. On the other hand
using too small a region means that the algorithm will miss the opportunity to take a step
substantial enough to move it much closer to the solution.
Each k-th step in the trust region algorithm is obtained by solving the sub-problem
defined by:
min
dk
mkdk=fxk+dT
k∇fxk+1
2dT
k∇2fxkdk,dk≤k(8)
where kis the trust region radius. By writing the unknown direction as a linear combi-
nation of Newton and steepest descent direction, the sub-problem will take the following
form:
min mkxk=fxk+s1dSD
k+s2dN
kT∇fxk
+1
2s1dSD
k+s2dN
kT∇2fpks1dSD
k+s2dN
k,(9)
under the constrains:
s1dSD
k+s2dN
k
≤k.(10)
The problem now becomes two-dimensional and it is solved for the unknown coeffi-
cients s1and s2. In order to find both s1and s2in Equation (10) a set of non-linear equations
can be solved using, for example, the Newton-Raphson techniques.
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INVERSE PROBLEMS IN SCIENCE AND ENGINEERING 7
As was mentioned in the introductory part of this section the MLBA is a minimization
procedure performed on two separate levels. In an internal level the discrepancy between
measured and computed quantities is minimized for a fixed value of h1. In other words, the
optimisation algorithm for a given initial value of hi=1
1finds a solution, in terms of three
elastic moduli: Ei=1
1,Ei=1
2,Ei=1
3. The procedure converges when the mean square error
reaches its minimum value (RMSE →min). This condition, in the classical implementa-
tion of inverse analysis, is sufficient to complete the task. According to the authors, however,
the analysis should proceed with a different values of hi
1in order to find possibly better
solution through a multilevel nested analysis. Therefore, in a next step, the optimization
algorithm draws another first layer’s thickness hi
1and searches for a set of elastic moduli
values satisfying the condition RMSEi+1< RMSEi. So the converged solution of each
internal minimizations brings an objective function value of a single external iteration. On
each level, the same minimization algorithm, namely TRA is implemented for discrepancy
minimization between computed and measured data (here numerically generated in order
to mimic the dynamic FWD measurement test). In order to make an algorithm more
sensitive to RMSE, changes in the new concept of deflection slope derivatives will be used
herein and briefly explained in the next section.
2.3. Deflection slope derivatives
Most of the methods associated with the backcalculation of pavement’s elastic modulus
are related to the direct comparison of deflections curves. Here the new concept is studied
which uses the deflection curve’s first derivatives ∂u/∂r. Such transformation of deflection
curve helps to increase the sensitivity of RMSE with respect to the model parameters.
Deflections curve is transformed into its first derivative using the forward, central and
backward second order of accuracy finite difference in the first point (i=1), all midpoints
(i=2...6) and the last one (i=7), respectively, by the following formulas:
∂u
∂r
ri=1
=−3ui+4ui+1−ui+2
2h(11a)
∂u
∂r
ri=2...6
=−ui−1+ui+1
2h(11b)
∂u
∂r
ri=7
=ui−2−4ui−1+3ui
2h(11c)
The above assumption is an obvious simplification, which introduces a significant error
in the derivative estimate of a central point, where its value should be 0 due to axial
symmetry of deflection curve. Also derivatives in the vicinity of axis of symmetry are
incorrect, however, authors deliberate choice of derivatives (curvatures) computations
based on discrete points located on one side of symmetry plane only gives very good
indication of models’ differences.
Let’s consider two similar pavement structures with parameters shown in the Table 1.
In such a case the RMSE of the two deflections curves has a relatively small value. The
difference between both curves is rather small. If one takes the transformed curves, several
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8 T. GARBOWSKI AND A. PO ˙
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Table 2. Parameters and backcalculation results in comparison to referential values (A and B Models).
Case 11 Case 22
Start Converged values Reference Start Converged values Reference
Parameters values displ. deriv. Model A values displ. deriv. Model B
E1[MPa] 10.000 7.026 7.000 7.000 10.000 9.434 9.426 9.368
E2[MPa] 200 297 300 300 200 297 301 300
E3[MPa] 70 100 100 100 70 100 99 100
h1[m] 0.20 – – 0.20 0.18 – – 0.18
h2[m] 0.30 – – 0.30 0.30 – – 0.30
times increase in the difference in terms of RMSE measurement is observed. The effect of
such curve transformation and its influence on RMSE is visualized in Figure 2.
The deflection curves of two pavement models shown in Figure 2a practically overlap
though the elastic moduli of first layer which differ more than 33%. Such circumstances
reveal that if thickness isn’t correctly measured, the determined elastic modulus can be
far different from the real value. In contrary, the deflection line curvature is much more
sensitive to model differences and therefore can be more effectively used in thickness
estimate correction. Obviously since the difference occurs in the first layer only the most
visible curvature deviation are present in the close vicinity of deflection marked in a centre
of the load.
3. Numerical results
3.1. Single-level backcalculation approach
Since the correct values of each elastic modulus are known a-priori, therefore to present
the convergence process all computed quantities in each iteration are here divided by their
reference values. Thus, if sought parameters converge to one during the minimization
process, it means they converge to the exact values. Figures 3and 4show the convergence
of the minimization procedure based on displacement curves (Figures 3(a) and 4(a), and
on derivatives of vertical deflections (Figures 3(b) and 4(b). In all examples, thickness of
the first layer was set to the exact value. It meets the standard pavement backcalculation
procedure called herein a single-level backcalculation approach (SLBA).
Table 2shows detailed converged values of sought parameters in examples given in the
Figures 3and 4. The calculations take into consideration a deflection curve with white
noise. The value range of the noise is comprised in ±1µm.
In contrary, Figures 5and 6show the convergence process of different models in
which the thickness h1was set to a wrong value. Here, the elastic modulus identified
through an inverse analysis has significantly different values than reference ones due to a
compensation phenomenon. Setting thickness to lower then reference value automatically
generates higher stiffness in order to retain the vertical deflections of the model surface.
Table 3shows the detailed converged values of sought parameters in examples given in
the Figures 5and 6. The level of noise of the deflection curves in the pavement model was
consistently assumed to be ±1µm.
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INVERSE PROBLEMS IN SCIENCE AND ENGINEERING 9
Figure 2. Diagrams for A and B models: (a) the shape of deflection curves, (b) the shape of the curve drawn on the basis of the derivatives of deflection.
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10 T. GARBOWSKI AND A. PO ˙
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Figure 3. Convergence of the model A parameters (case 11): (a) displacements, (b) derivatives.
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INVERSE PROBLEMS IN SCIENCE AND ENGINEERING 11
Figure 4. Convergence of the model B parameters (case 22): (a) displacements, (b) derivatives.
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12 T. GARBOWSKI AND A. PO ˙
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Figure 5. Convergence of the model A parameters (case 12): (a) displacements, (b) derivatives.
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INVERSE PROBLEMS IN SCIENCE AND ENGINEERING 13
Figure 6. Convergence of the model B parameters (case 21): (a) displacements, (b) derivatives.
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14 T. GARBOWSKI AND A. PO ˙
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Table 3. Parameters and backcalculation results in comparison to referential values (A and B models).
Case 12 Case 21
Start Converged values Reference Start Converged values Reference
Parameters values displ. deriv. Model A values displ. deriv. Model B
E1[MPa] 10.000 8.340 8.061 7.000 10.000 7.760 8.164 9.368
E2[MPa] 200 362 388 300 200 245 199 300
E3[MPa] 70 100 98 100 70 101 106 100
h1[m] 0.18 – – 0.20 0.20 – – 0.18
h2[m] 0.30 – – 0.30 0.30 – – 0.30
Table 4. Parameters and backcalculation results in comparison to referential values (A and B models).
Case 12 Case 21
Start Converged values Reference Start Converged values Reference
Parameters values displ. deriv. Model A values displ. deriv. Model B
E1[MPa] 15.000 6.973 6.863 7.000 15.000 8.777 8.941 9.368
E2[MPa] 500 281 299 300 500 258 290 300
E3[MPa] 50 100 100 100 50 103 101 100
h1[m] 0.18 0.2019 0.2015 0.2000 0.20 0.1880 0.1839 0.1800
h2[m] 0.30 – – 0.30 0.30 – – 0.30
3.2. Multi-level backcalculation approach
Herein, thickness of the first layer model is subjected to a minimization procedure in an
external loop, whereas elastic moduli are identified in subsequent internal loops. In each
external iteration, a full minimization problem concerning elastic moduli characterization
is solved for a fixed value of thickness h1. Figures 7and 8show the convergence of both:
(a) thickness h1and (b) elastic moduli Ei,i=1...3.
The convergence of h1is observed here within 2–3 iteration (external loop), while elastic
moduli Eiare properly determined within 4–5 iterations (internal loop) in each external
iteration. In order to speed up the computation the trust region algorithm implemented
in an internal minimization loop uses the converged values from the previous external
iteration so the convergence curves of elastic moduli depicted in both Figures 7and 8form
a continuous lines, whereas the h1curves have characteristic discontinuities. This jumps
point out the external iterations.
Table 4gather all converged values of sought parameters in these both examples in
where the MLBA was applied. Again the noisy data were used (±1µm).
3.3. Discussion
When the typical pavement backcalculation procedure is used, the thickness of each
layer is assumed as a constant value, therefore the elastic moduli of individual layers
are the only unknown parameters to be determined. An inverse analysis performed in
such a framework, fits an algorithmic scheme called here the single-level backcalculation
approach, which, if tuned with wrong pavement thickness values, converge to the wrong
stiffness parameters. The study outlined in this work shows that incorrectly assumed
thickness of the first pavement layer (different about two centimetres from its correct value)
has a significant influence on the output of the inverse algorithm. The backcalculated values
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INVERSE PROBLEMS IN SCIENCE AND ENGINEERING 15
Figure 7. Convergence of the model A parameters (case 12): (a) deflections and (b) derivatives.
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16 T. GARBOWSKI AND A. PO ˙
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Figure 8. Convergence of the model B parameters (case 21): (a) deflections, (b) derivatives.
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INVERSE PROBLEMS IN SCIENCE AND ENGINEERING 17
of the elastic moduli are, in this case, substantially under or overestimated, and the RMS
value reaches even 30%. This observation applies to both presented here inverse procedure:
(a) based on deflection curves and (b) on its derivatives.
The correct values of the sought parameters can be obtained by introducing the proposed
here procedure, which uses a multi-level backcalculation algorithm. So both elastic moduli
of the pavement layers and the thickness of the first layer are simultaneously investigated.
Even though the thickness of the first layer again has been introduced incorrectly (differing
by a two centimetres from its correct value), the algorithm precisely converged to the
sought parameters (RMS < 1%). The subsequent steps in the external loop iteratively
updates the thickness h1and therefore improves the estimates of elastic moduli E1,E2and
E3in the internal loop. This procedure provides the solution which is considerably more
accurate compared to the SLBA. Even better results can be achieved when the derivatives
of the deflection curves are used instead of deflection basins itself. Comparing the results
achieved basing on the derivatives of the deflection curves, the converged solution is 5%
more accurate then corresponding solution computed using deflection curves only.
Analysis and general performance of the proposed multi-level backcalculation algo-
rithm presented here are based on the synthetically generated data. This approach is
very often applied to the preliminary testing stage of the new algorithms, mainly because
both the sought and model parameters are known a priori and therefore the procedure
can be easily checked and confronted. Only knowing precisely each ingredient of the
model, i.e. thickness of each layer, its stiffness and Poisson’s ratios, material temperature,
interlayer bounds, etc. one can controllably generate pseudo-experimental results (here
surface deflections). In the real experimental set-up, it is rather difficult to eliminate all the
unknown influences and therefore is rarely used for testing purposes.
4. Conclusions
Knowing the sensitivities of the measurable quantities with respect to the model param-
eters, one can increase the number of sought parameters to be investigated in an inverse
procedure. However, releasing more parameters makes the problem ill-conditioned. The
remedy is an iterative procedure updating some parameters in the internal loop while
keeping others fixed in the external loop for the subsequent steps. The approach presented
here uses this methodology, thus is capable to determine not only the elastic moduli but
also the thickness of the asphalt layer.
Based on the selected pseudo-experimental examples of the three-layer pavement mod-
els, the novel inverse procedure is proposed here and preliminary tested. The procedure is
based on the minimization of the discrepancy between the reference pseudo-experimental
(additionally noised) and computed deflection curves. It is noteworthy that an oscillation
of measurement’s uncertainties on the level of few micrometers is considered as a noise
which can heavily affects the backcalculation results, thus here the noise level ±1µmisused,
which provides stable results. This is an important message showing that while measuring
pavement deflections in-situ, one should use a method ensuring the high precision.
In order to identify both the stiffnesses and thicknesses of the pavement model, a new
formulation of the multi-level backcalculation algorithm is proposed here.
The proposed algorithm automatically includes the natural changes in the layers’
thickness along the uniform section that results from the construction inaccuracies and
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18 T. GARBOWSKI AND A. PO ˙
ZARYCKI
defects of the used technology. In the present practice, the in situ destructive test is used
to identify the local thickness which is later extrapolated to a uniform section of the road
(often hundreds of meters). Therefore the use of the standard backcalculation procedure
based on assumed thicknesses, cen results with an error reaching 30%.
In the proposed inverse procedure, the derivatives of the deflection curves are used. This
particular construction of the objective function is tested and compared to the traditional
implementation based on the deflection curves only. In the case of the standard single-level
backcalculation approach, the transformation of the deflection curves does not improve
the solution. On the other hand, using the new multi-level backcalculation algorithm
produces results that are by a several per cent more precise. For example, providing that
the starting value of h1(asphalt concrete thickness) in the optimization algorithm does
not differ from the real one by more than 2 centimetres, the results of the inverse analysis
based on the derivatives of deflection curves are 5% more precise than the results achieved
by a standard procedure. Taking into account the number of factors affecting the inverse
results, the obtained here improvements bring us closer to the more realistic and robust
identification of pavement parameters.
Disclosure statement
No potential conflict of interest was reported by the authors.
Funding
This work has been partially supported by the NCBiR research [grant number PBS3/B6/38/2015].
ORCID
Tomasz Garbowski http://orcid.org/0000-0002-9588-2514
Andrzej Po˙zarycki http://orcid.org/0000-0002-1321-066X
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