Julien Berger
Pontifical Catholic University of Paraná, Brazil

Denys Dutykh

arXiv:1610.03688v1 [physics.comp-ph] 12 Oct 2016

CNRS–LAMA, Université Savoie Mont Blanc, France

Nathan Mendes
Pontifical Catholic University of Paraná, Brazil

On the optimal experimental
design for heat and moisture
parameter estimation

arXiv.org / hal

Last modified: October 13, 2016

On the optimal experimental design for heat
and moisture parameter estimation
Julien Berger∗ , Denys Dutykh, and Nathan Mendes

Abstract. In the context of estimating material properties of porous walls based on
in-site measurements and identification method, this paper presents the concept of Optimal Experiment Design (OED). It aims at searching the best experimental conditions in
terms of quantity and position of sensors and boundary conditions imposed to the material.
These optimal conditions ensure to provide the maximum accuracy of the identification
method and thus the estimated parameters. The search of the OED is done by using the
Fisher information matrix and a priori knowledge of the parameters. The methodology is
applied for two case studies. The first one deals with purely conductive heat transfer. The
concept of optimal experiment design is detailed and verified with 100 inverse problems
for different experiment designs. The second case study combines a strong coupling between heat and moisture transfer through a porous building material. The methodology
presented is based on a scientific formalism for efficient planning of experimental work
that can be extended to the optimal design of experiments related to other problems in
thermal and fluid sciences.
Key words and phrases: inverse problem; parameter estimation; Optimal Experiment
Design (OED); heat and moisture transfer; sensitivity functions
MSC: [2010] 35R30 (primary), 35K05, 80A20, 65M32 (secondary)
PACS: [2010] 44.05.+e (primary), 44.10.+i, 02.60.Cb, 02.70.Bf (secondary)

Key words and phrases. inverse problem; parameter estimation; Optimal Experiment Design (OED);
heat and moisture transfer; sensitivity functions.
∗
Corresponding author.

Contents
1

Introduction

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2

Optimal Experiment Design for non-linear heat transfer problem . . . . . . . 5

2.1 Physical problem and mathematical formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Optimal experiment design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 Numerical example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Estimation of one single parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Verification of the ODE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3

Optimal Experiment Design for a non-linear coupled heat and moisture transfer
problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.1 Physical problem and mathematical formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2 Optimal experiment design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.3 Numerical example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Estimation of one single parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Estimation of several parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

A

Hygrothermal properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

J. Berger, D. Dutykh & N. Mendes

4 / 31

1. Introduction
Heating or cooling strategies for buildings are commonly based on numerical building
physics mathematical models, which can be calibrated using on-site measurements for
estimating the properties of the materials constituting the walls, reducing the discrepancies
between model predictions and real performance.
Several experimental works at the scale of the wall can be reported from the literature.
Instrumented test cells, as ones presented in [8, 9, 13, 14, 18, 22, 25, 27] provide measured
dataset as temperature and relative humidity at different points in the wall for given
boundary conditions. Some experiments at the scale of the whole-building are described
in [7, 17]. These data can be used to estimate the properties (transport and capacity
coefficients) of the materials as reported for instance in [19, 29] for heat transfer and in
[23, 34] for coupled heat and moisture transfer.
The estimation of the unknown parameters P, e.g. wall thermophysical properties, based
on observed data and identification methods is illustrated in Figure 1. Observed data are
experimentally obtained. The latter is established by a design defining the boundary
and initial conditions of the material, the type of sensors as well as their quantity and
location. Thus, the accuracy of the estimated parameters P strongly depends on the
experiment design. The choice of the measurement devices, of the boundary and initial
conditions have consequences on the estimation of the parameter. Furthermore, due to
the correlation between the parameters, multiple local solutions of the estimation problem
exist. Hence, one can address the following questions: what are the best experimental
conditions to provide the best conditioning of the identification method? In particular,
how many sensors are necessary? Where are their best locations? What boundary and
initial conditions should be imposed? Can we really choose them?
These issues deal with searching the Optimal Experiment Design (OED) that enables
to identify the parameters P of a model with a maximum precision. A first objective of
the OED is to adjust the conditions of the experiment design in order to maximize the
sensitivity of the output field u to parameters P. A second objective is to find the optimum
location and quantity of sensors. The achievement of these objectives enables to determine
conditions of the experiments under which the identification of the parameters will have
the maximum accuracy.
The search of the OED is based on quantifying the amount of information contained by
the observed field uexp . For this, Fisher information matrix is used [1, 2, 10, 11, 15, 26,
31, 32], considering both the model sensitivity, the measurement devices and the parameter
correlation. The sensitivity of the output field u with respect to the model parameters P is
calculated, corresponding to the sensitivity of the cost function of the parameter estimation
problem. The higher the sensitivity is, the more information is available in the measurement
data and the more accurate is the identification of the parameters. Generally speaking, the
methodology of searching the ODE is important before starting any experiment aiming at
solving parameter estimation problems. It allows choosing with a deterministic approach

On the optimal experimental design

5 / 31

the conditions of the experiments. Furthermore, it provides information on the accuracy
of parameter estimation.
Several works can be reported on the investigation of OED [1–3, 10, 11, 15, 26, 31, 32].
Among them, in [3], the OED is planned as a function of the quantity of sensors and their
location, for the estimation of boundary heat flux of non-linear heat transfer. In [15], the
OED is searched for the estimation of transport coefficient in convection-diffusion equation.
In [20], the OED is analysed as a function of the boundary heat flux for the identification
of the radiative properties of the material. In [30], the OED is investigated for chemical
reactions using a Bayesian approach. However, the application of OED theory for nonlinear heat and moisture transfer with application for the estimation of the properties of a
wall is relatively rare.
This article presents the methodology of searching the OED for experiments aiming
at solving parameter estimation problems. In the first Section, the concept of OED is
detailed and verified for an inverse problem of non-linear heat transfer. The computation
of the model sensitivity to the parameter is specified. The OED is sought as a function
of the quantity and location of sensors as well as the amplitude and the frequency of the
heat flux at the material boundaries. Then the OED for the estimation of hygrothermal
properties considering non-linear heat and moisture transfer is investigated. Finally, some
main conclusions and perspectives are outlined in the last Section.

2. Optimal Experiment Design for non-linear heat transfer
problem

First, the methodology of searching the OED is detailed for a problem of non-linear heat
transfer. A brief numerical example is given to illustrate the results.

2.1. Physical problem and mathematical formulation
The physical problem considers an experiment of a one-dimensional transient conduction
heat transfer in domains x ∈ Ω = [0, 1] and t ∈ [0, τ ]. The initial temperature in the
body is supposed uniform. The surface of the boundary ΓD = {x = 1} is maintained at
the temperature uD . A time-harmonic heat flux q, of amplitude A and frequency ω, is
imposed at the surface of the body denoted by Γq {x = 0}. Therefore, the mathematical

6 / 31

J. Berger, D. Dutykh & N. Mendes

Boundary conditions
Initial conditions

Model
Direct problem
Sensitivity problem

Searching the OED
Fisher information matrix
D-optimum criteria Ψ

OED
π◦

Experiments
Data

Parameter estimation problem

Sensors location X
Sensors quantity N
Sensors errors σ 2

Identification methods

parameter P◦
Figure 1. Process of searching the Optimal Experiment Design.

formulation of the heat conduction problem can be written as:



∂u
∂
∂
⋆
c ⋆ − ⋆ k (u) ⋆ u = 0
∂t
∂x
∂x
u = uD
∂u
k ⋆ (u) ⋆ = A⋆ sin(2πω ⋆ t⋆ )
∂x
u = u0 (x⋆ )
⋆

⋆

k (u) =

(k0⋆

+

k1⋆ u)

x⋆ ∈ Ω, t⋆ ∈ ]0, τ ] ,

(2.1a)

x⋆ ∈ ΓD , t > 0,

(2.1b)

x⋆ ∈ Γq , t > 0,

(2.1c)

x⋆ ∈ Ω, t⋆ = 0,

(2.1d)
(2.1e)

7 / 31

On the optimal experimental design

where the following dimensionless quantities are introduced:
x⋆ =

x
,
L

u=

k1⋆ =

k1 Tref
,
kref

c⋆ =

T
,
Tref
c
cref

,

uD =

TD
,
Tref

u0 =

T0
,
Tref

k0⋆ =

k0
,
kr

tref =

cref L2
,
kref

A⋆ =

AL
,
kref Tref

ω ⋆ = ωtref

where T is the temperature, c the volumetric heat capacity, k0 the thermal conductivity
and k1 its dependency on temperature, L the linear dimension of the material, A the
intensity and ω the frequency of the flux imposed on surface Γq . Subscripts ref accounts
for a characteristic reference value, D for the Dirichlet boundary conditions, zero (0) for
the initial condition of the problem and superscript ⋆ for dimensionless parameters.
The problem given by Eqs. (2.1)(a-e) is a direct problem when all the thermophysical
properties, initial and boundary conditions, as well as the body geometry are known. Here,
we focus on the inverse problem consisting in estimating one or more parameters of the
material properties (as c, k0 and/or k1 ) using the mathematical model Eqs. (2.1)(a-e)
and a measured temperature data uexp obtained by N sensors placed in the material at
X = [xn ] , n ∈ {1, . . . , N}. The M unknown parameters are here denoted as vector P =
[pm ] , m ∈ {1, . . . , M}. Here, the inverse problems are all non-linear, over-determined
solved by considering a least-square single-objective function:
J [P] = ||uexp − T (u (x, t, P)) ||2

(2.2)

where u is the solution of the transient heat conduction problem Eqs. (2.1)(a-e). uexp
are the data obtained by experiments. They are obtained by N sensors providing a time
discrete measure of the field at specified points within the material. T is the operator
allowing to compare the solution u at the same space-time points where observations uexp
are taken. The cost function can be defined in its matrix form, using the ordinary least
squares norm, as:

⊤ 

J [P] = Uexp − U (P) Uexp − U (P)

where U is the vector containing the discrete point of the field u for the discrete set of
space-time points obtained by the experiments.
For this study, several experiments are distinguished. Ones aim at estimating one parameter: m = 1 and P = c, P = k0 or P = k1 . Others aim at estimating a group of m = 2
or m = 3 parameters. In this work, the optimal experiment design will be investigated for
both cases.

2.2. Optimal experiment design
Efficient computational algorithms for recovering parameters P have already been proposed. Readers may refer to [21] for a primary overview of different methods. They are
based on the minimisation of the cost function J [P]. For this, it is required to equate to

8 / 31

J. Berger, D. Dutykh & N. Mendes

zero the derivatives of J [P] with respect to each of the unknown parameters pm . Associated to this necessary condition for the minimisation of J [P], the scaled dimensionless
local sensitivity function [12] is introduced:
Θm (x, t) =

σp ∂u
,
σu ∂pm

∀m ∈ {1, . . . , M}

(2.3)

where σu is the variance of the error measuring uexp . The parameter scaling factor σp equals
1 as we consider that prior information on parameter pm has low accuracy. It is important
to note that all the algorithm have been developed considering the dimensionless problem
in order to compare only the order of variation of parameters and observation (and thus
avoid units and scales effects).
The sensitivity function Θm measures the sensitivity of the estimated field u with respect
to change in the parameter pm [3, 20, 21]. A small value of the magnitude of Θm indicates
that large changes in pm yield small changes in u. The estimation of parameter pm is
therefore difficult in such case. When the sensitivity coefficient Θm is small, the inverse
problem is ill-conditioned. If the sensitivity coefficients are linearly dependent, the inverse
problem is also ill-conditioned. Therefore, to get an optimal evaluation of parameters P,
it is desirable to have linearly-independent sensitivity functions Θm with large magnitude
for all parameters pm . These conditions ensure the best conditioning of the computational
algorithm to solve the inverse problem and thus the better accuracy of the estimated
parameter.
It is possible to define the experimental design in order to reach these conditions. The
issue is to find the optimal quantity of sensors N ◦ , their optimal location X◦ , the optimal
amplitude A◦ and the optimal frequency ω ◦ of the flux imposed at the surface Γq . To
search this optimal experiment design, we introduce the following measurement plan:
(2.4)

π = {N, X, A, ω}

In analysis of optimal experiment for estimating the unknown parameter(s) P, a quality
index describing the accuracy of recovering is the D−optimum criterion [1–4, 10, 11, 15,
20, 26, 31, 32]:
(2.5)

Ψ = det [F (π)]
where F (π) is the normalized Fisher information matrix [15, 31]:
F (π) = [Φij ]
N ˆ τ
X
Φij =
Θi (xn , t)Θj (xn , t)dt
n=1

, ∀(i, j) ∈ {1, . . . , M}2

(2.6a)
(2.6b)

0

The matrix F (π) characterises the total sensitivity of the system as a function of measurement plan π Eqs. (2.4). The search of the OED aims at finding a measurement plan
π ⋆ for which the objective function Eq. (2.5) reaches the maximum value:
π ◦ = {N ◦ , X◦ , A◦ , ω ◦ } = arg max Ψ
π

(2.7)

On the optimal experimental design

9 / 31

To solve problem (2.7), a domain of variation Ω π is considered for the quantity of sensors
N, their location X, the amplitude A and the frequency of the flux. Then, the following
steps are done for each value of the measurement plan π = {N, X, A, ω} in domain Ω π .
The direct problem defined byEqs. (2.1)(a-e) is computed. In this work, it is solved by
using a finite-difference standard discretization. An embedded adaptive in time Runge–
Kutta scheme combined with central spatial discretization is used. It is adaptive and
embedded to estimate local error with little extra cost.
Given the solution u of the direct problem 2.1(a-e) for a fixed value of the measurement
∂u
by solving the sensitivity problem
plan, the next step consists in computing Θm = ∂p
m
associated to parameter pm :

 

∂
∂k
∂k ∂ 2 u
∂Θm
∂
∂c ∂u ∂u ∂
k Θm = −
+
c
−
+
x ∈ Ω, t > 0,
∂t
∂x
∂x
∂pm ∂t
∂x ∂pm ∂x
∂pm ∂xx
Θm = 0
x ∈ Γd , t > 0,
∂u ∂k
∂Θm
x ∈ Γq , t > 0,
=
k
∂x
∂x ∂pm
Θm = 0
x ∈ Ω, t = 0,
k = (k0 + k1 u)

(2.8a)
(2.8b)
(2.8c)
(2.8d)
(2.8e)

The sensitivity problem Eqs. (2.8) is also solved using an embedded adaptive in time
Runge–Kutta scheme combined with central spatial discretization.
It is important to note that the solution of the direct (2.1) problem and the sensitivity
(2.8)(a-e) problem are done for a given parameter P. The later is chosen with the prior
information. The validity of the OED depends on the a priori knowledge. If there is
no prior information, the methodology of the OED can be done using an outer loop on
the parameter P sampled using Latin hypercube or Halton quasi-random samplings for
instance.
Then, given the sensitivity coefficients, the Fisher matrix (2.6)(a,b) and the D−optimum
criterion (2.5) are calculated.

2.3. Numerical example
Current section illustrates the search of optimum experiment design for problem Eqs.
(2.1) considering u0 = uD = 1. The dimension-less properties of the material are equal
to c⋆ = 10.9, k0⋆ = 1, k1⋆ = 0.12. The final simulation time of the experiment is fixed to
τ = 28.3.
From a physical point of view, the numerical values correspond to a material of length
Lr = 0.1 m. The thermal properties were chosen from a wood fibre: c = 3.92 · 105 J/m3 /K,
k0 = 0.118 W/m/K and k1 = 5 · 10−4 W/m/K2 . The initial temperature of the material
is considered uniform at T0 = Tref = 293.15 K. The temperature at the boundary ΓD is
maintained at TD = Tref = 293.15 K. The characteristic time of the problem is tref = 3050

10 / 31

J. Berger, D. Dutykh & N. Mendes

(a)

(b)

Figure 2. Schematic view of experimental design (a) and the extreme values of
the heat flux (b).

s. Thus, the total time of the experiments corresponds to 24 h. A schematic view of the
problem is given in Figure 2a.
As mentioned in previous section, the OED is sought as a function of the quantity of
sensors N, their location X, the amplitude A and frequency ω of the flux imposed at the
surface Γq . For the current numerical application, we consider a maximum of N = 4 possible sensors. Their location varies from X = [0] for N = 1 and X = [0 0.25 0.5 0.75] for
N = 4 as shown in Figure 2a. The variance of the error measurement equals σT = 0.05◦C.
For the amplitude A, 5 values are considered in the interval [0.2, 1] with a regular mesh
of 0.2. The minimal value of A corresponds to a physical heat flux of 70 W/m2 . For the
frequency, 30 values have been taken in the interval [10−3, 1]. The extreme values of the
heat flux are illustrated in Figure 2b.
2.3.1 Estimation of one single parameter
Considering previous description of the numerical example, we first focus on the search
of the OED for designing an experiment aiming at estimating one parameter. Thus, P
equals to c, k0 or k1 . Figures 3a, 3b and 3c show the variation of the criterion Ψ as a
function of the amplitude A and the frequency ω of the heat flux. For each of the three
experiments, the criterion is maximum for a low frequency and for a large amplitude as
presented in Figure 6. Numerical values of the OED π ◦ are given in Table 1. In terms of
physical numerical values, the OED is reached for a heat flux of amplitude 350 W/m2 and
a period of 17.3 h, 60.6h and 53.5h for estimating parameter c, k0 or k1 , respectively.

On the optimal experimental design

11 / 31

The effect of the numbers of sensors and their positions is given in Figures 4a, 4b and
4c for a fixed amplitude. The criterion increases with the number N of sensor. Adding
new sensors yields to a more optimum design. However, it can be noticed the slope of the
increase has a small magnitude. For a single sensor N = 1, almost 95% of the maximal
criterion is reached. Therefore, if the amount of sensors for the experiments is limited, just
one sensor located at the boundary receiving the heat flux is available reasonable, enabling
to recover one parameter (c, k0 or k1 ). Indeed, the boundary Γq is where the sensitivity
coefficient of the parameter has the largest magnitude as shown in Figures 5a, 5b and 5c
correspondingly.
It can be noticed in Figure 5d that the sensitivity coefficients on the surface Γq ≡ {x = 0}
are linearly-independent. Therefore, the inverse problem will not be very sensitive to
measurement errors which enables an accurate estimation of the parameters P = (c, k0 , k1 ).
The OED is based on the solution of the direct problem 2.1 and the sensitivity problem 2.8, computed for a given value of the parameter P. The a-priori knowledge on
parameter P is crucial for the success of the OED. For some applications, there is no prior
information on the parameter. For these cases, it is possible to operate an outer loop on
the parameter P sampled in a given domain. For this numerical example, it is considered
that there is no prior information on parameters k0 and k1 . It is only known their values belong to the domains [0.1, 0.2] W/m/K and [1 10] 10−4 W/m/K2 . The a priori value of the
volumetric heat capacity is still fixed to c = 3.92 · 105 J/m3 /K. A Halton quasi-random
algorithm [16] is then used to generate a sampling of 100 couples of parameters (k0 , k1 )
in the domains, as illustrated in Figure 7a. For each couple (k0 , k1 ), the methodology described in Section 2.2 is performed to compute the OED. Figure 7b gives the variation of
optimal frequency ω ◦ with parameter k0 and k1 . The blue points correspond to the results
of the OED method for each couple of the discrete sampling. A surface of variation of
ω ◦ is then interpolated. It can be noted that the increase of ω ◦ is more important with
parameter k1 .
2.3.2 Verification of the ODE
In order to illustrate the robustness of the ODE, several inverse problems are solved, considering different measurement designs. We consider 30 inverse problems (2.1) of parameter
P associated to the 30 values of the frequency ω in the interval [10−3, 1]. For a fixed value
of ω, N e = 100 experiments were designed to recover the parameters P equals to c, k0
or k1 and to P = (c, k0, k1 ). For this purpose, the simulated discrete time observation is
obtained by numerical resolution of the direct problem and a uniform sampling with time
period ∆t = 10 min. A normal distribution with a standard deviation of σ = 0.01 was
assumed. The inverse problem is solved using Levenberg-Marquardt method [6, 21, 33].
After the solution of the N e × 30 inverse problems, the empirical mean square error is
computed:
r
1
(P − P ◦ (ω)) 2 ,
(2.9)
EMSE (ω) =
Ne

12 / 31

J. Berger, D. Dutykh & N. Mendes

(a) IP(c)

(b) IP(k0 )

(c) IP(k1 )

(d) IP((c, k0 , k1 ))

Figure 3. D−optimum criterion Ψ as a function of A and ω for the 4 different
experiments (N = N ◦ = 1).

where P ◦ is the estimated parameter by the resolution of the inverse problem.
The procedure is then repeated for the quantity N and location X of the sensors of the
measurement plans and a fixed value of the frequency ω = ω ◦ . The empirical error is
computed with equation (2.9) as a function of N (as P now depends on N). This approach
is also applied for parameters P equals to c, k0 or k1 and P = (c, k0 , k1 ) .
The evolution of the empirical mean square error is illustrated in Figures 8a and 8b for
the parameter estimation problem corresponding to the four experiments.

13 / 31

On the optimal experimental design

(a)IP(c)

(b)IP(k0 )

(c)IP(k1 )

(d)IP(c, k0 , k1 )

Figure 4. D−optimum criterion Ψ as a function of N , X and ω for the 4
different experiments (A = A◦ = 1).

In Figure 8a, the variation of the error with the number of the sensors is smooth for the
experiments for the estimation of a single parameter. However, for the estimation of several
parameters P = (c, k0 , k1 ) , the EMSE is below 10−2 for three sensors. These results are
in accordance with the variation of the criterion Ψ, in Figure 5d. An important variation
of the magnitude of Ψ can be observed when passing from X = 0 to X = [0 0.25].
In Figure 8b, the error is minimised for the optimum experiment design ω ◦ for the four
experiments. The estimation of the parameter tends to be inaccurate when the frequency

14 / 31

J. Berger, D. Dutykh & N. Mendes

(a) IP(c)

(b) IP(k0 )

(c) IP(k1 )

(d) IP(c, k0 , k1 )

Figure 5. Sensitivity coefficient of parameters for the four different experiments
(N = N ◦ = 1).

tends to zero, as the heat flux tends to zero and there is almost no solicitation of the
material.
For the experiments for the estimation of parameter P = c, the error variation is smooth.
As illustrated in Figure 8b, the peak of the criterion Ψ has a large deviation. Therefore,
if the frequency ω of the experiments is different from the one of the ODE ω ◦ , the error
to estimate the parameter might still be acceptable. For the other experiments, we can
notice that the peak of the criterion Ψ has a small deviation in Figures 5b, 5c and 5d.

15 / 31

On the optimal experimental design

Figure 6. Optimal heat flux for the four different experiments (N = 1).

(a)

(b)

Figure 7. Halton quasi-random sampling of parameters (k0 , k1 ) (a) and
frequency optimal experiment design ω ◦ (b).

16 / 31

J. Berger, D. Dutykh & N. Mendes

Unknown parameter

Optimal experimental design π ◦

max(Ψ)
A◦ (-)

A◦
(W/m2 )

1
ω◦

(-)

1
ω◦

(h)

N◦

X◦

P=c

1.4 ×
10−2

1

350

20.4

17.3

1

0

P = k0

8.6

1

350

71.6

60.6

1

0

P = k1

3.23

1

350

63.2

53.5

1

0

P = (c, k0 , k1 )

9.1 ×
10−3

1

350

29.7

25.2

3

0

Table 1. Value of the maximum D−optimum criterion for the four different
experiments.

(a)

(b)

Figure 8. Evolution of the empirical mean square error with the number of
sensors (a) and with the frequency ω (b).

The peak of the criterion Ψ has a small deviation in Figures 5b, 5c and 5d. Consequently,
the variation of the error in Figure 8b is more important for estimating parameters P = k0 ,
P = k1 or P = (c, k0 , k1 ) . If the frequency of the experiment design is different from
the one of the OED, it implies a loose of accuracy in the estimation of these parameters.

17 / 31

On the optimal experimental design

3. Optimal Experiment Design for a non-linear coupled
heat and moisture transfer problem
In the previous section, the concept of optimal experiment design was detailed for a
non-linear heat transfer problem. It has also been verified for 100 inverse problems for
different experiment designs by calculating the error as a function of the frequency and the
number of sensors of the measurement plan. In next section, the approach goes further
by studying optimal experiment design to estimate transport and storage coefficients of a
heat and moisture transfer.

3.1. Physical problem and mathematical formulation
The physical problem concerns a 1-dimensional coupled heat and moisture transfer
through a wall based on [5, 23, 24, 28]:




∂T
∂
∂Pv
∂
∂T
d1
− (Lv − cl T )
d2
=0
−
c10
∂t
∂x
∂x
∂x
∂x


∂w
∂Pv
∂
d2
=0
−
∂t
∂x
∂x

(3.1a)
(3.1b)

with w the water content, Pv the vapour pressure, T the temperature, d2 the vapour permeability, c10 the volumetric thermal capacity, d1 the thermal conductivity, cl the specific
heat capacity of water and Lv the latent heat of evaporation. As this study remains in the
hygroscopic range of the material, the liquid transport is not presently taken into account.
The following assumptions are adopted on the properties of the material. The volumetric
moisture content is assumed as a first-degree polynomial of the vapour pressure. The
vapour permeability and the thermal conductivity are taken as a first-degree polynomial
of the volumetric moisture content:
w
c21
= c20 +
Pv
w0
w0
w
d2 = d20 + d21
w0
w
k = d10 + d11
w0

(3.2a)
(3.2b)
(3.2c)

Based on these equations, the experimental set-up considers a material with uniform
initial temperature and vapour pressure. At t > 0, sinusoidal heat and vapour flux are
imposed at boundary Γq = {x = 0}, while the temperature and vapour pressure are maintained constant at the other boundary ΓD = {x = 1}. The unscaled problem can be

18 / 31

J. Berger, D. Dutykh & N. Mendes

formulated as:




∂
⋆ ∂u
⋆ ∂v
d1 ⋆ − (Ko1 − Ko2 ) Lu ⋆ d2 ⋆ = 0 x⋆ ∈ Ω, t⋆ ∈ ]0, τ ]
∂x
∂x
∂x

c⋆10

∂u
∂
− ⋆
⋆
∂t
∂x

c⋆21

∂v
∂
− Lu ⋆
⋆
∂t
∂x



⋆ ∂v
d2 ⋆ = 0
∂x

(3.3a)

x⋆ ∈ Ω, t⋆ ∈ ]0, τ ]

(3.3b)

− d⋆1

∂u
= A⋆1 sin(2πω1⋆t⋆ )
⋆
∂x

x⋆ ∈ Γq , t⋆ ∈ ]0, τ ]

(3.3c)

− d⋆2

∂v
= A⋆2 sin(2πω2⋆t⋆ )
∂x⋆

x⋆ ∈ Γq , t⋆ ∈ ]0, τ ]

(3.3d)

u = uD ,
u = u0 (x),
d⋆1
d⋆2

=
=

x⋆ ∈ ΓD , t⋆ ∈ ]0, τ ] (3.3e)

v = vD

d⋆10
d⋆20

+
+

x⋆ ∈ Ω, t⋆ = 0

v = v0 (x)
d⋆11
d⋆21

(c20 +
(c20 +

c⋆21 v)
c⋆21 v)

(3.3f)
(3.3g)
(3.3h)

with the following dimension-less ratios:
u=

T
Tref

v=

Pv
Pref

u0 =

T0
Tref

uD =

TD
Tref

vD =

Pv,D
Pref

Ko1 =

Lu =

d2,ref c1,ref
c2,ref d1,ref

d⋆10 =

d10
d1,ref

d⋆11 =

d11
d1,ref

d⋆21 =

d21
d2,ref

c⋆10 =

c10

c⋆21 =

c21

A⋆1 =

A1 L
d1,ref Tref

A⋆2 =

A2 L
d2,ref Pref

ω1⋆ = ω1 tref

t⋆ =

t

x⋆ =

x
L

tref =

tref

c1,ref

Lv c2,ref Pref
c1,ref Tref

c2,ref

v0 =

Pv,0
Pref

Ko2 =

d⋆20 =

cL c2,ref
Pref
c1,ref
d20
d2,ref

c2,ref =

w0
Pref

ω2⋆ = ω2 tref

c1,ref L2
d1,ref

where Ko is the Kossovitch number, Lu stands for the Luikov number, L is the dimension of the material, tref , the characteristic time of the problem, A and ω the amplitude
and intensity of the heat and vapour fluxes. Subscripts ref accounts for a reference value,

19 / 31

On the optimal experimental design

D for the Dirichlet boundary conditions, 0 for the initial condition of the problem, 1 for
the heat transfer, 2 for the vapour transfer and superscript ⋆ for dimensionless parameters.

3.2. Optimal experiment design
The OED is sought as a function of the quantity of sensors N and their locations X
and as a function of the frequencies (ω1 , ω2 ) of the heat and vapour fluxes. According
to the results of Section 2.3 and to our numerical investigations, a monotonous increase
of the sensitivity of the system were observed with the amplitude (A1 , A2 ) of the flux.
Therefore, these parameters were considered as fixed. Thus, the OED aims at finding the
measurement plan π ◦ for which the criterion Eqs. (2.5) reaches a maximum value:
π ◦ = {N ◦ , X◦, ω1◦ , ω2◦} = arg max Ψ
π

(3.4)

Parameters Lv , cL are physical constants given for the problem. Therefore, considering
Eqs. (3.3), a number of 7 parameters can be estimated by the resolution of inverse problems:
(c⋆10 , d⋆10 , d⋆11 , c⋆20 , c⋆21 , d⋆20 , d⋆21 ). One can focus on the definition of an experiment for the
estimation of one single parameter or several parameters. It might be noted that parameters
(c⋆20 , c⋆21 , d⋆20 , d⋆21 ) can be identified by inverse problems considering field u, v or both (u, v)
as observation. The thermal properties (c⋆10 , d⋆10 , d⋆11 ) can only be estimated using the
observation of u.
All in all, 20 experiments can be defined as:
i.
ii.
iii.
iv.

15 for the estimation of single parameters among c⋆10 , d⋆10 , d⋆11 , c⋆20 , c⋆21 , d⋆20 or d⋆21 ,
1 for the estimation of the thermal properties (c⋆10 , d⋆10 , d⋆11 ),
3 for the estimation of the moisture properties (c⋆20 , c⋆21 , d⋆20 , d⋆21 ),
1 for the estimation of the hygrothermal properties (hg) (c⋆10 , d⋆10 , d⋆11 , c⋆20 , c⋆21 , d⋆20 , d⋆21 ).

Following notation is adopted: IP(p) [u] states for an experiment defined for the estimation
of parameter p using field u as the observation. The 20 experiments are recalled in Table 2.
The same methodology as presented in Section 2 is used. The fifteen sensitivity functions
are computed for calculating the criterion (3.4).

3.3. Numerical example
The following numerical values are considered for numerical application. The domain Ω
is defined as Ω = [0, 1], considering the wall thickness of the material as the characteristic
length of the problem Lr = 0.1m. The total simulation time of the experiments is τ =
6 × 103 , corresponding to a physical simulation of 40 days. The initial and prescribed
conditions equal to uD = u0 = 1 and vD = v0 = 0.5. The reference temperature and
vapour pressure are taken as Tref = 293.15K and Pv,ref = 2337Pa, respectively. The
amplitude of the heat and vapour fluxes are A⋆1 = 1.7 × 10−2 and A⋆2 = 1.7, equivalent to
600W/m2 and 1.2 × 10−7kg/m3 /s.

20 / 31

J. Berger, D. Dutykh & N. Mendes

The dimension-less parameters are:
Lu = 2.5 × 10−4 Ko1 = 2.1 × 10−1 Ko2 = 2.5 × 10−2 d⋆10 = 5 × 10−2 d⋆11 = 5 × 10−3
d⋆20 = 1

d⋆21 = 0.4

c⋆10 = 1

c⋆20 = 2

c⋆21 = 6

The properties correspond to a wood fibre material [5, 23]. They are given in its physical
dimension in Appendix A.
The OED is sought as a function of the number of the sensors N. It varies from N = 1,
located at X = [0], to N = 3, located at X = [0 0.2 0.4]. The variances of the
measurement error are σT = 0.05◦ and σP = 2Pa. The OED is also investigated as a
function of the frequencies (ω1⋆, ω2⋆ ) of the flux. For each frequency, 20 values are taken in
the interval [1 × 10−5 ; 1.5 × 10−3 ]. The minimal and maximal values correspond to a flux
having a physical period of 495 days and 3.3 days, respectively.
As mentioned in Section 2.2, the computation of the solution of the optimal experiment
plan is done by successive iterations for the whole grid of the measurement plan π =
{N, X, ω1 , ω2 } .
3.3.1 Estimation of one single parameter
In the current section, the OED is sought for experiments to estimate one single parameter among c⋆10 , d⋆10 , d⋆11 , c⋆20 , c⋆21 , d⋆20 or d⋆21 . The results of the ODE are given in Table 2
for the physical values and in Figure 11b for the dimensionless values.
The criterion Ψ varies actively with the frequencies (ω1⋆ , ω2⋆ ) for estimating parameter
c⋆10 , as shown in Figure 9a. The ODE is reached for a period for the heat and vapour flux
of 27.2 days.
On the other hand, Figures 9b and 9c illustrate that the criterion varies mostly with the
frequency ω1⋆ for estimating parameter d⋆10 and d⋆11 . The ODE is reached for a period for the
heat of 78.1 days. Furthermore, the magnitude of Ψ is really higher than zero, ensuring a
good conditioning to solve inverse problems. As observed in the previous section concerning
a non-linear heat transfer problem (Section 2.3), the ODE period of the heat flux is shorter
for the thermal capacity than for the thermal conductivity.
In Figure 12, the sensitivity functions of parameters d⋆10 is given for experimental conditions π where Ψ reaches its minimal value and for the ODE conditions (where Ψ reaches
it maximal value). In Figure 12a, the magnitude of the sensitivity function is almost 50
times smaller than one for the ODE conditions (Figure 12b). Therefore, the estimation
of the parameters might be less accurate for this conditions than the other ones. In Figure 12b, it can be also noticed that the sensitivity function is maximal at the boundary
Γq = {x = 0} . It emphasizes why the criterion Ψ is maximal for a single sensor settled
at this boundary.
For experiments estimating the vapour properties, c⋆20 , c⋆21 , d⋆20 or d⋆21 , the ODE is not
very sensitive to the frequency of the heat flux as reported in Figures 9d, 9e, 9f and 10a.
It can be noted that the criterion Ψ is higher when dealing with experiments considering
fields (u, v) as observations. The computational algorithm to solve the inverse problem is

On the optimal experimental design

21 / 31

better conditioned. The period of the ODE vapour flux is 9.5 and 12.3 days for experiments
estimating d⋆20 and c⋆21 . Experiments for parameters d⋆21 and c⋆20 have the same period of
the ODE vapour flux (27.2 days).
It might be recalled that this analysis has been done for a fixed and constant error measurement, equals for the temperature and vapour pressure sensors. Indeed, this hypothesis
can be revisited in practical applications. Furthermore, if only one field is available to
estimate the vapour properties (u or v), it is required to use the field v as observation and
prioritize the accuracy for those sensors. The criterion Φ and the sensitivity is highest for
the field v as shown in Figures 13a and 13b.
For all experiments, a single sensor located at x = 0 is sufficient to reach more than 95%
of the maximum criterion as given in Table 2. The surface receiving the heat and vapour
flux is where the sensitivity of the parameters is the higher as illustrated in Figures 13a
and 13b for the parameter c21 .
3.3.2 Estimation of several parameters
The optimal experiment design is now sought for experiments to estimate several parameters: the thermophysical properties (c⋆10 , d⋆10 , d⋆11 ) , the moisture properties (c⋆20 , c⋆21 , d⋆20 , d⋆21 )
and the hygrothermal (hg) coefficients (c⋆10 , d⋆10 , d⋆11 , c⋆20 , c⋆21 , d⋆20 , d⋆21 ) . Five experiments
are considered for the estimation of these parameters as reported in Table 2.
For the estimation of the hygrothermal properties, Figure 10b shows that the criterion Ψ
varies mostly with frequency ω1⋆ . The criterion is very close to zero (an order of O(10−9 )).
The computational algorithm for the solution of the inverse problem might be ill conditioned. The results of the inverse problem might not be accurate. A way to circumvent
the problem is to increase the precision of the sensors and the amplitude of the heat and
vapour fluxes.
According to Table 2, the moisture properties might be estimated using both fields (u, v) .
If it is not possible, the field v would give a more accurate estimation than field u. The
criterion varies mostly with the frequency ω1⋆ of the heat flux, Figure 10c. The ODE is
reached for a period of 35.4 and 16 days for the heat and vapour fluxes, respectively.
For the thermal properties, the criterion varies with both heat and vapour flux frequencies, as reported in Figure 10d. A period of 16 days for both fluxes yields the ODE.
The variation of the criterion with the quantity of sensors is given in Figure 14. As
expected, the criterion increases with the number of sensors for all experiments. For the
estimation of the hygrothermal properties, the ODE is achieved for 3 sensors. As the vapour
properties might be estimated using both fields (u, v), the use of two sensors, located at
x = 0 and x = 0.2m, for each field, is a reasonable choice that enables to reach more
than 95% of the criterion Ψ . In the case, for any reasons, the measurement of both fields is
not possible, three sensors measuring the field v are required for the ODE. For the thermal
properties, the use of only two sensors is reasonable, as 95% of the maximum criterion is
reached. These results are synthesised in Table 2.

22 / 31

J. Berger, D. Dutykh & N. Mendes

Optimal experimental design π ⋆
2π
2π
◦
ω1
ω2◦
N◦
(days)
(days)

Unknown parameter

max {Ψ}

IP(c10 )[u]

6.03

27.2

27.2

1

IP(d10 )[u]

288

78.1

20.9

1

IP(d11 )[u]

181

78.1

101.7

1

IP(d20 )[u]

0.53

7.3

7.3

1

IP(d20 )[v]

0.99

60

9.5

1

IP(d20 )[u, v]

1.53

9.5

9.5

1

IP(d21 )[u]

133

78.1

27.2

1

IP(d21 )[v]

140

9.5

27.2

1

IP(d21 )[u, v]

276

78.1

27.2

1

IP(c20 )[u]

0.02

20.9

20.9

1

IP(c20 )[v]

0.03

9.5

27.2

1

IP(c20 )[u, v]

0.05

27.2

27.2

1

IP(c21 )[u]

0.004

35.4

20.9

1

IP(c21 )[v]

0.014

78.1

12.3

1

IP(c21 )[u, v]

0.017

78.1

12.3

1

IP(hg)[u, v]

4.5 ×
10−9

27.2

12.3

3

IP(c20 , c21 , d20 , d21 )[u]

0.001

60.0

20.9

3

IP(c20 , c21 , d20 , d21 )[v]

0.15

12.3

27.2

3

IP(c20 , c21 , d20 , d21 )[u, v]

0.2

35.4

16

2

IP(c10 , d10 , d11 )[u]

137

16

16

2

Table 2. Value of the maximum D−optimum criterion for each experiments.

23 / 31

On the optimal experimental design

(a) IP(c10 )[u], N = 1

(b) IP(d10 )[u], N = 1

(c) IP(d11 )[u], N = 1

(d) IP(d20 )[u,v], N = 1

(e) IP(d21 )[u,v], N = 1

(f) IP(c20 )[u,v], N = 1

Figure 9. D−optimum criterion Ψ as a function of the frequencies (ω1 , ω2 ) .

24 / 31

J. Berger, D. Dutykh & N. Mendes

(a) IPc21 [u,v], N = 1

(b) IP(hg)[u,v], N = 2

(c) IP(c20 , c21 , d20 , d21 )[u,v], N = 1

(d) IP(c10 , d10 , d11 )[u], N = 1

Figure 10. D−optimum criterion Ψ as a function of the frequencies (ω1 , ω2 ) .

4. Conclusions
In the context of estimating material properties, using in-site measurements of wall in
test cells or real buildings combined with identification methods, this study explored the
concept of optimal experiment design (OED). It aimed at searching the best experimental
conditions in terms of quantity and location of sensors and flux imposed to the material.
These conditions ensure to provide the best accuracy of the identification method and thus

25 / 31

On the optimal experimental design

(a)

(b)

Figure 11. Frequencies (ω1⋆◦ , ω2⋆◦ ) of the ODE.

(a)

(b)

Figure 12. Sensitivity coefficient of parameter k0 for experimental conditions π
where Ψ reaches its minimal value (a) ( ω2π1 = 3.3 days, ω2π2 = 495 days, N = 3)
and for the ODE conditions (b) ( ω2π◦ = 78.1 days, ω2π◦ = 20.9 days, N ◦ = 1).
1

2

the estimated parameter. The search of the OED was done using the Fisher information
matrix, quantifying the amount of information contained in the observed field.

26 / 31

J. Berger, D. Dutykh & N. Mendes

(a)

(b)

Figure 13. Sensitivity coefficient of parameter c21 for the ODE conditions of
IP(c21 )[u, v] .

Figure 14. D−optimum criterion Ψ as a function of the quantity of sensors N .

On the optimal experimental design

27 / 31

Two cases were illustrated. The first one dealt with an inverse problem of non-linear
heat transfer to estimate the thermal properties (storage and transport coefficients), considering a uniform initial temperature field and, as boundary conditions, a fixed prescribed
temperature on one side and a sinusoidal heat flux on the other one, for 24 hours. The
ODE yields in using one single temperature sensor located at the surface receiving the
flux. The flux should have an intensity of 350 W/m2 and experiment periods of 17 h, 61 h,
54 h and 25 h for the estimation of thermal capacity, thermal conductivity, temperature
dependent conductivity and all parameters, respectively. For this case study, the concept
of optimal experiment was verified by solving 100 inverse problems for different experiment
designs and calculating the error as a function of the frequency and the number of sensors
of the measurement plan. It has been demonstrated that the accuracy of the parameter is
higher when parameters are recovered with measurements carried out via ODE.
The second application concerned experiments for inverse problems of a coupled nonlinear heat and moisture transfer problem to estimate the hygrothermal properties of the
material. The experiment is similar to the first case study. Uniform initial distribution of
temperature and vapour pressure fields were considered, submitted to harmonic heat and
vapour fluxes at one side and prescribed temperature and vapour pressure values at the
other one. The experiment was done for a period of 40 days. The achievement of the ODE
was explored for different experiments, aiming at estimating one or several parameters. As
the equation considered are weakly coupled, the thermal properties can only be determined
using the temperature. The accuracy of the identification method does not depend on the
vapour flux. For the vapour properties, results have shown that the estimation will be
more accurate using the temperature and vapour pressure as observations. Furthermore,
the accuracy actively depends on the period of the vapour flux. Single sensor has to
be located at the side where the flux is imposed. For experiments to estimate all the
hygrothermal properties, two sensors are enough to improve the accuracy.
This contribution explored the concept of optimal experiment design for application in
building physics for estimating the hygrothermal properties of construction materials. The
methodology of searching the ODE is important before starting any experiment aiming at
solving parameter estimation problems. With a priori values of the unknown parameters,
the sensitivity functions and the optimum criterion can be computed. Results allow choosing by means of deterministic approach the conditions of the experiments. A good design
of experiments avoids installing unnecessary sensors. In the case of coupled phenomena,
as the combined heat and moisture transfer problem, considering sensor accuracies, ODE
enables to choose and select the field that must be monitored. It also improves the accuracy
of the solution of the estimation problem.
Further work is expected to be carried out with different design strategies (ODE and
others), estimating properties using real observations.

28 / 31

J. Berger, D. Dutykh & N. Mendes

Properties

Value

d10 (W/m/K)

0.5

d11 (W/m/K/Pa)

0.05

d20 (s)

2.5 × 10−11

d21 (s/Pa)

1 × 10−11

c11 (J/m3 /K)

4 × 105

c20 (-)

2

c21 (s2 /m2 )

2.5 × 10−2

Physical constant

Value

Lv (J/kg)

2.5 × 106

cL (J/kg)

1000

Table 3. Hygrothermal properties of the material.

Acknowledgments
The authors acknowledge the Brazilian Agencies CAPES of the Ministry of Education and
the CNPQ of the Ministry of Science, Technology and Innovation for the financial support.
Dr. Dutykh also acknowledges the hospitality of PUCPR during his visit in April 2016.

A. Hygrothermal properties
The hygrothermal properties of the material used in Section 3.3 are given in Table 3.

References
[1] O. M. Alifanov, E. A. Artioukhine, and S. V. Rumyantsev. Extreme Methods for Solving
Ill-Posed Problems with Applications to Inverse Heat Transfer Problems. Begellhouse, New
York, 1995. 4, 5, 8
[2] M. L. Anderson, W. Bangerth, and G. F. Carey. Analysis of parameter sensitivity and
experimental design for a class of nonlinear partial differential equations. Int. J. Num. Meth.
Fluids, 48(6):583–605, jun 2005. 4

On the optimal experimental design

29 / 31

[3] E. A. Artyukhin and S. A. Budnik. Optimal planning of measurements in numerical experiment determination of the characteristics of a heat flux. Journal of Engineering Physics,
49(6):1453–1458, dec 1985. 5, 8
[4] J. V. Beck and K. J. Arnold. Parameter Estimation in Engineering and Science. John Wiley
& Sons, Inc., New York, 1977. 8
[5] J. Berger, M. Chhay, S. Guernouti, and M. Woloszyn. Proper generalized decomposition for
solving coupled heat and moisture transfer. Journal of Building Performance Simulation,
8(5):295–311, sep 2015. 17, 20
[6] J. Berger, S. Gasparin, M. Chhay, and N. Mendes. Estimation of temperature-dependent
thermal conductivity using proper generalised decomposition for building energy management. Journal of Building Physics, jun 2016. 11
[7] R. Cantin, J. Burgholzer, G. Guarracino, B. Moujalled, S. Tamelikecht, and B. Royet. Field
assessment of thermal behaviour of historical dwellings in France. Building and Environment,
45(2):473–484, feb 2010. 4
[8] T. Colinart, D. Lelievre, and P. Glouannec. Experimental and numerical analysis of the
transient hygrothermal behavior of multilayered hemp concrete wall. Energy and Buildings,
112:1–11, jan 2016. 4
[9] T. Z. Desta, J. Langmans, and S. Roels. Experimental data set for validation of heat, air
and moisture transport models of building envelopes. Building and Environment, 46(5):1038–
1046, may 2011. 4
[10] A. F. Emery and A. V. Nenarokomov. Optimal experiment design. Measurement Science
and Technology, 9(6):864–876, jun 1998. 4, 5, 8
[11] T. D. Fadale, A. V. Nenarokomov, and A. F. Emery. Two Approaches to Optimal Sensor
Locations. Journal of Heat Transfer, 117(2):373, 1995. 4, 5, 8
[12] S. Finsterle. Practical notes on local data-worth analysis. Water Resources Research,
51(12):9904–9924, dec 2015. 8
[13] C. James, C. J. Simonson, P. Talukdar, and S. Roels. Numerical and experimental data set
for benchmarking hygroscopic buffering models. Int. J. Heat Mass Transfer, 53(19-20):3638–
3654, sep 2010. 4
[14] T. Kalamees and J. Vinha. Hygrothermal calculations and laboratory tests on timber-framed
wall structures. Building and Environment, 38(5):689–697, may 2003. 4
[15] M. Karalashvili, W. Marquardt, and A. Mhamdi. Optimal experimental design for identification of transport coefficient models in convection-diffusion equations. Computers & Chemical
Engineering, 80:101–113, sep 2015. 4, 5, 8
[16] L. Kocis and W. J. Whiten. Computational investigations of low-discrepancy sequences.
ACM Transactions on Mathematical Software, 23(2):266–294, jun 1997. 11
[17] M. Labat, M. Woloszyn, G. Garnier, and J. J. Roux. Dynamic coupling between vapour
and heat transfer in wall assemblies: Analysis of measurements achieved under real climate.
Building and Environment, 87:129–141, may 2015. 4
[18] D. Lelievre, T. Colinart, and P. Glouannec. Hygrothermal behavior of bio-based building
materials including hysteresis effects: Experimental and numerical analyses. Energy and
Buildings, 84:617–627, dec 2014. 4
[19] A. Nassiopoulos and F. Bourquin. On-Site Building Walls Characterization. Numerical Heat
Transfer, Part A: Applications, 63(3):179–200, jan 2013. 4

J. Berger, D. Dutykh & N. Mendes

30 / 31

[20] A. V. Nenarokomov and D. V. Titov. Optimal experiment design to estimate the radiative
properties of materials. Journal of Quantitative Spectroscopy and Radiative Transfer, 93(13):313–323, jun 2005. 5, 8
[21] M. N. Ozisik and H. R. B. Orlande. Inverse Heat Transfer: Fundamentals and Applications.
CRC Press, New York, 2000. 7, 8, 11
[22] H. Rafidiarison, R. Rémond, and E. Mougel. Dataset for validating 1-D heat and mass
transfer models within building walls with hygroscopic materials. Building and Environment,
89:356–368, jul 2015. 4
[23] S. Rouchier, M. Woloszyn, Y. Kedowide, and T. Béjat. Identification of the hygrothermal
properties of a building envelope material by the covariance matrix adaptation evolution
strategy. Journal of Building Performance Simulation, 9(1):101–114, jan 2016. 4, 17, 20
[24] H.-J. Steeman, M. Van Belleghem, A. Janssens, and M. De Paepe. Coupled simulation
of heat and moisture transport in air and porous materials for the assessment of moisture
related damage. Building and Environment, 44(10):2176–2184, oct 2009. 17
[25] E. Stéphan, R. Cantin, A. Caucheteux, S. Tasca-Guernouti, and P. Michel. Experimental
assessment of thermal inertia in insulated and non-insulated old limestone buildings. Building
and Environment, 80:241–248, oct 2014. 4
[26] N.-Z. Sun. Structure reduction and robust experimental design for distributed parameter
identification. Inverse Problems, 21(2):739–758, apr 2005. 4, 5, 8
[27] P. Talukdar, S. O. Olutmayin, O. F. Osanyintola, and C. J. Simonson. An experimental
data set for benchmarking 1-D, transient heat and moisture transfer models of hygroscopic
building materials. Part I: Experimental facility and material property data. Int. J. Heat
Mass Transfer, 50(23-24):4527–4539, nov 2007. 4
[28] F. Tariku, K. Kumaran, and P. Fazio. Transient model for coupled heat, air and moisture
transfer through multilayered porous media. Int. J. Heat Mass Transfer, 53(15-16):3035–
3044, jul 2010. 17
[29] S. Tasca-Guernouti, B. Flament, L. Bourru, J. Burgholzer, A. Kindinis, R. Cantin, B. Moujalled, G. Guarracino, and T. Marchal. Experimental Method to determine Thermal Conductivity, and Capacity Values in Traditional Buildings. In VI Mediterranean Congress of
Climatization, Madrid, Spain, 2011. 4
[30] G. Terejanu, R. R. Upadhyay, and K. Miki. Bayesian experimental design for the active
nitridation of graphite by atomic nitrogen. Experimental Thermal and Fluid Science, 36:178–
193, jan 2012. 5
[31] D. Ucinski. Optimal Measurement Methods for Distributed Parameter System Identification.
2004. 4, 5, 8
[32] A. Vande Wouwer, N. Point, S. Porteman, and M. Remy. An approach to the selection
of optimal sensor locations in distributed parameter systems. Journal of Process Control,
10(4):291–300, aug 2000. 4, 5, 8
[33] C.-Y. Yang. Estimation of the temperature-dependent thermal conductivity in inverse heat
conduction problems. Appl. Math. Model., 23(6):469–478, jun 1999. 11
[34] A. Zaknoune, P. Glouannec, and P. Salagnac. Estimation of moisture transport coefficients in
porous materials using experimental drying kinetics. Heat and Mass Transfer, 48(2):205–215,
feb 2012. 4

On the optimal experimental design

31 / 31

Thermal Systems Laboratory, Mechanical Engineering Graduate Program, Pontifical
Catholic University of Paraná, Rua Imaculada Conceição, 1155, CEP: 80215-901, Curitiba
– Paraná, Brazil
E-mail address: Julien.Berger@pucpr.edu.br
URL: https://www.researchgate.net/profile/Julien_Berger3/
LAMA, UMR 5127 CNRS, Université Savoie Mont Blanc, Campus Scientifique, 73376 Le
Bourget-du-Lac Cedex, France
E-mail address: Denys.Dutykh@univ-savoie.fr
URL: http://www.denys-dutykh.com/
Thermal Systems Laboratory, Mechanical Engineering Graduate Program, Pontifical
Catholic University of Paraná, Rua Imaculada Conceição, 1155, CEP: 80215-901, Curitiba
– Paraná, Brazil
E-mail address: Nathan.Mendes@pucpr.edu.br
URL: https://www.researchgate.net/profile/Nathan_Mendes/

