Mathematical Analysis of Two Approaches for Optimal Parameter Estimates to Modeling Time Dependent Properties of Viscoelastic Materials ()
1. Introduction
In this paper, we assume that X is the space of all continuous functions
having a Laplace transform
with
.
Parameters
associates with a time-domain model
are considered optimal insofar as they yield a minimal model-observation discrepancy
defined by
, where function
is obtained as a regression to a set of time-dependent observations. The model-observation discrepancy
is assumed to be function-valued, so the phrase “minimal discrepancy” only has meaning when
is understood to be a member of some norm-induced topology
. Having specified the norm-induced topology to which
belongs, the optimal parameters are then computed as an optimal solution
to the least squares problem (LSP)
(1)
Two norms on X are considered in formulating the LSP (1).
The first norm, the baseline norm, is denoted by
, while the second norm, the alternative norm, is denoted by
. (The norms
and
on X are defined in Section 2.) The use of the baseline norm
in (1) yields a variant of a commonly used LSP for computing optimal model parameters, while the alternative norm
is motivated by the elegant closed-form expressions for certain models
undertaking the Laplace transform. This is particularly true for certain creep models associated with viscoelastic materials [1] - [7].
While the use of the alternative norm
in LSP (1) has been successfully applied for computing optimal parameter estimates in [5], a theoretical foundation and justification for the use of the alternative form
in LSP (1) is in need of further development. Refining the developments began in [8] [9], this paper addresses the above need in Section 2, where 1) two inner products
and
are defined over X and verified with respect to the inner product properties; 2) the norms
and
are induced from the respective inner products
and
; 3) from the inner product properties, a bounding relationship is established between the norms
and
; and 4) insight is obtained from the bounding relationship into how the parameter solutions
to LSP (1)
relate to the parameter solutions
to LSP (1)
. The first three contributions represent a substantial refinement and streamlining of the developments in [8] [9], thus paving the way for the fourth contribution which, furthermore, builds on the developments in [8] [9].
The remainder of the paper is organized as follows. From the developments in Section 2, a more simple and improved implementation of a previous application [5] becomes evident, and this is presented in Section 3. Computational setup and results are presented and discussed briefly in this same section. Lastly Section 4 concludes this paper and provides comments on future work.
2. Definition and Analysis of the Norms
and
The two norms
and
are induced, respectively, by the following two inner products
and
defined in the following manner for each pair
and parameters
and
:
(2)
(3)
It is now shown that (2) and (3) are, in fact, inner products.
Proposition 2.1. The mappings
given by (2) and
given by (3) are defined for all
and are furthermore inner products over X.
Proof: Because X contains the continuous functions
having a Laplace transform, the inner product
is defined for all
. Also, the function fg defined by multiplying
and
is continuous and of exponential order [10] (follows from the same properties of f and g), and so the Laplace transform
exists, and (2) is simply the Laplace transform
evaluated at
. Thus, the inner product
is also defined for all
.
Recall that, for any vector space V, an inner product
satisfies the following rules for each
and
(e.g., see [11]):
I1:
I2:
I3:
I4:
, and
Property I1 follows readily for
by noting that f and g are real-valued functions
is real-values, and so the integrand is real-valued. For
, Property I1 follows from (3) by computing
where F(s) and G(s) denote the Laplace transform of f and g, respectively.
Properties I2 and I3 follow easily for both
and
from elementary properties of integrals.
Property I4 applies to
because: 1) for each
, the integrand of
is always nonnegative; and 2) if
, then by the continuity of f over
, there exist
,
, and
over which
for all
. Thus, for each
, we have
if
. From this, the implication
follows.
To show that property I4 applies to
, first note that
for all
follows from the definition (3), and so it remains to show that
. This latter claim holds under application of Lerch’s theorem (see, e.g., [11] [12]) to the setting where f is continuous. Namely, if
(so that
), then
for all
. The assumed continuity of f on
and the Fundamental Theorem of Calculus imply that
. Thus, I4 holds for
. Hence, it has been shown that
and
are both inner products over X.
One possible relationship between two different norms
and
called equivalence is now explored. The equivalence of two norms
and
is characterized by the existence of
such that
for all
(4)
(See, e.g., [11].) Using the inner-product structures defined on X, the Cauchy-Schwartz inequality can be used to show a bounding relationship of the form
for all
,
, and
via the computation
(5)
(6)
where
.
Whereas the upper bound coefficient u is established in (6), the lower bound coefficient
necessary to establish the equivalence (4) for each fixed
and
is shown not to exist through two counterexamples:
Counterexample 1: Let f be of the form
. Then
and
. So
. Both
and
. Furthermore, since
, there is no
serving as a lower bound coefficient.
Counterexample 2: Let f be of the form
,
. Then
and
. Now
and
. Thus,
, and so there is no lower bound
on
.
The lack of a lower bound coefficient
is also depicted in Figure 1 and Figure 2 for the same two counterexamples. Thus, it is established that due to the lack of the lower bound coefficient
, the norms
and
over X are not equivalent.
The bounding relationship (6) between the norms
and
is also described via inclusion relationships between sublevel sets. The sublevel set
is defined by
Figure 1. Illustrating the lack of a lower bound coefficient
for the norms
(
) and
(
) with
. For both plots, each point corresponds to the use of a single value of
, where
.
Figure 2. Plot of points
with
,
,
, and frequency parameter
varying from
to
. The plotted points approaching the origin along the plotted curve correspond to
values approaching zero, while the plotted points proceeding away from the origin along the same plotted curve correspond to
values approaching infinity.
for each f, P, and
. By the existence of the bounding coefficient
, in (6), we have the inclusion
(7)
The sublevel set inclusion (7) provides a sense in which the norm
penalizes model-observation discrepancy more leniently than the norm
. This leniency is observed, for example, in the plot of Figure 2 where the increasing frequency of
due to
leads to
while
.
For application purposes, the preference between the norms
and
in formulating the LSP (1) depends on 1) the desired degree of leniency in penalizing imperfect model-observation fit due to the use of parameter
; and 2) the ease and accuracy of evaluating the norms
and
. Next, in Section 3, the material science application of solving LSP (1) motivating the contributions of this paper is revisited where the use of each of the two norms
and
is evaluated in terms of the above two preference criteria.
3. Application for Modeling Time Dependent Properties of Viscoelastic Materials
A time-dependent model
for modeling creep of viscoelastic materials under an applied stress load is given by
(8)
where the stress level σ and Young’s modulus E are determined experimentally, and the material-specific kernel parameters
satisfy
(See [1] [3] [5] for details.) The parameter
can be found from the first term of the infinite series expansion in (8) [3]. Thus, only the model parameters
and
need to be determined as an optimal solution
to problem (1) with
.
The regression function
is fit to observations based on experiments performed for three types of composites with nanofillers [5]:
1) Pure polyamide (PA).
2) Polyamide with ultra-dispersed diamonds (PA + UDD).
3) Polyamide with carbon nanotube fillers (PA + CNT).
For each material, the tests with the corresponding three loading levels
,
, and
are performed, where the subscript of
indicates that the stress applied to the materials is 30%, 40%, and 50%, respectively, of the ultimate stress, which was taken equivalent to the yielding stress of each of the tested materials. Using these experimental data, the regression functions
used for each data set take the form
(9)
where the coefficients
are estimated for each data set using standard linear regression techniques. The resulting regression functions and the material-specific vales for
,
, and
are given in Table 1.
For each computation, the norm
parameter
and the norm
parameter
are used; furthermore, the experimentally determined parameters α, E, and
associated with
are provided in Table 2.
Table 1. Regression functions obtained from the creep experiments.
The optimal parameters
are computed as optimal solutions to LSP (1) using the baseline norm
and the alternative norm
. These computations are performed with MapleTM [13]. The computed parameter estimates are presented in Table 3 and the resulting wellness-of-fit between the parameterized models and experimental observations are illustrated in Figure 3.
As observed earlier [1] [3], the model
has an elegant simplification under its Laplace transformation
(10)
Furthermore, each function r with the form (9) has a closed-form Laplace transform denoted by
. Thus, for each s satisfying
, problem (1) takes the following elegant form when
:
(11)
Solving the LSP (11) is computationally more accurate and less expensive than solving the corresponding LSP (1) with
. This is consistent with the
Figure 3. Wellness of fit plots using optimal parameter
solutions to problem (1) with
(left) and
(right). Plots are given based on nine experimental data sets corresponding to three materials each with three loading levels.
Table 3. Optimal parameter estimates.
motivation and observation seen in earlier works [1] [3] [14] associated with the use of Laplace transform-based approaches to estimating the optimal model parameters.
4. Conclusions
This paper contributes a mathematical foundation for the comparison between time domain least squares parameter estimation problems formulated using the norm
and Laplace domain least squares parameter estimation problems introduced in [1] [3], applied in [5] [8], and formulated using the alternative norm
as defined in Section 2. A relationship between the norms
and
is analyzed in terms of norm equivalence, and in exploring this equivalence, the existence of the necessary upper bound coefficient
was shown to exist in Section 2 using the two inner product structures (2) and (3) defined on X. However, the non-existence of the corresponding lower bound coefficient
, is demonstrated through two counterexamples. From the bounding relationship (6), inclusion relationships (7) of sublevel sets follow that provides a sense in which the norm
penalizes certain types of model-observation deviation more leniently than the norm
.
The plots of Figure 3 suggest that the solutions
to LSP (1) with
yield improved model-observation fit over the corresponding solutions with
. In addition to the computational advantages associated with solving (11), the improvement is also attributed to the relatively lenient (in a sense derived from the inclusion relationships (7)) penalization of certain types of model-observation by
as compared with
. If the types of model-observation deviations that are penalized leniently are subjectively negligible to the model user, then the computation of the optimal solutions
to LSP (1) with
is more flexible, and this results in subjectively improved model-observation fit as compared with the fit obtained with the use of the norm
.
Acknowledgements
The authors thank Mr. Brian Dandurand of Argon National Laboratory, Chicago, IL. For valuable insights, discussions, and computations provide.
The List of the Variables Used in This Paper
p: parameters in time-domain model
: model equation
: strain
: regression function
: norm induced topology
X: space of all condition functions of real variables
F: Laplace transformation
: baseline norm in real domain
: alternative norm in Laplace complex domain
V: vector space
u, v, w: vectors
λ: constant
s: complex variable
t: real variable
f, g: real valued functions
F(s), G(s): Laplace transforms of f andg functions
L: lower bound coefficient
ω: real parameter > 0
γ: complex valued parameter
δ: small real number
σ: stress level
E: Young’s modulus
α, β, λ: material specific kernel parameters
Γ: Gamma function
ci: regression function coefficients