PreprintPDF Available

Parametric PGD model used with orthogonal polynomials to assess efficiently the building's envelope thermal performance

Authors:
Preprints and early-stage research may not have been peer reviewed yet.

Abstract and Figures

Estimating the temperature field of a building envelope could be a time-consuming task. The use of a reduced-order method is then proposed: the Proper Generalized Decomposition method. The solution of the transient heat equation is then re-written as a function of its parameters: the boundary conditions, the initial condition, etc. To avoid a tremendous number of parameters, the initial condition is parameterized. This is usually done by using the Proper Orthogonal Decomposition method to provide an optimal basis. Building this basis requires data and a learning strategy. As an alternative, the use of orthogonal polynomials (Chebyshev, Legendre) is here proposed.
Content may be subject to copyright.
arXiv:2111.09102v1 [cs.CE] 17 Nov 2021
Parametric PGD model used with orthogonal polynomials to assess
efficiently the building’s envelope thermal performance
Marie-Hélène Azam1,2,, Julien Berger3, Sihem Guernouti1,2, Philippe Poullain1, Marjorie Musy1,2
November 18, 2021
1Université de Nantes, GeM UMR 6183 CNRS/Université de Nantes/Centrale Nantes, F-44600 Saint Nazaire, France
2Cerema, Equipe de Recherche BPE, F-44000 Nantes, France
3Université de La Rochelle, LaSIE UMR 7356 CNRS, F-17000, La Rochelle, France
Corresponding author: Azam Marie-Hélène, marie-helene.azam@univ-nantes.fr, IUT de Saint-Nazaire, Département
Génie Civil, 58 rue Michel Ange, F-44600 Saint Nazaire
Abstract: Estimating the temperature field of a building envelope could be a time-consuming task. The use of a
reduced-order method is then proposed: the Proper Generalized Decomposition method. The solution of the transient
heat equation is then re-written as a function of its parameters: the boundary conditions, the initial condition, etc. To
avoid a tremendous number of parameters, the initial condition is parameterized. This is usually done by using the Proper
Orthogonal Decomposition method to provide an optimal basis. Building this basis requires data and a learning strategy.
As an alternative, the use of orthogonal polynomials (Chebyshev, Legendre) is here proposed.
Key words : Heat transfer; Model order reduction methods; POD; PGD; Approximation Basis; Orthogonal Polyno-
mials.
1 Introduction
Modeling the thermal behavior of a building or a group of buildings is a challenging task. It implies that several
physical phenomena have to be taken into account: short and long-wave radiative heat balance, sensible and latent heat
flow transported by outdoor air movement and conductive heat transfer through the materials. All those heat fluxes may
vary over time and through space and thus lead to complex and non-uniform boundary conditions.
To quantify the global heat loss of a building envelope, the balance between the outdoor thermo-radiative heat fluxes
and the indoor ones must be estimated. One way of solving the global problem is to split the problem into several sub-
problems relative to (i) the outside thermo-radiative balance, (ii) the inside thermo-radiative balance, and (iii) the heat
transfer through the envelope [1]. Each problem is solved using a numerical model for each set of governing equations.
The whole energy model represents then the aggregation of those several sub-models through a coupling procedure also
called co-simulation [2]. Each numerical model exchanges parameters (i.e. surface energy balance or surface temperature
in the case of a thermal problem) with the other models during the simulation process.
This combination of models results in a large computation complexity and we need to reduce the computational times.
We focus here on the problem of the building envelope. To solve the global problem, for each element of the building
(walls, floor, etc.), the temperature field needs to be computed. For that purpose, the transient heat transfer equation needs
to be solved for the previously described boundary conditions (indoor and outdoor thermo-radiative balance). Usually, a
classical numerical model is then used based on finite difference, finite element, or finite volume. Those methods provide
an accurate solution but for a high computation cost.
To reduce the computational time keeping an accurate solution, the use of model order reduction methods is currently
investigated. The main idea is to replace the detailed and time-consuming model with a reduced-order model. For that
purpose, we investigate the use of the Proper Generalized Decomposition (PGD) method.
1
Parametric PGD model used with orthogonal polynomials to assess efficiently the building’s envelope thermal performance
Applied to urban soil heat transfer modeling, this model reduction method has shown its efficiency [3]. A cut com-
putational cost of 80% was observed for a mean surface temperature error below 0.52 C. Applied to building wall heat
transfer modeling, the PGD parametric model computes the solution 100 times faster than a classical numerical method
[4].
To reduce the numerical complexity of the problem, the solution is decomposed as a function of parameters like the
boundary conditions, or the initial condition. The efficiency of the PGD method relies on the number of parameters
used. To obtain a minimum number of parameters, some of them are usually combined through approximation. Those
approximations are done by the projection of the field of interest on an approximation basis.
Selecting the right approximation basis that will guarantee the model final accuracy with a minimum number of
parameters is a challenging task. The purpose of this article is then to overcome this obstacle. We investigate here the
use of a polynomial basis like Chebyshev or Legendre. Out of the approximation theory [5], those basis have proven
to be very efficient at solving partial differential equations using spectral methods [6,7].
The use of a polynomial basis is compared to the use of a classical reduced-order basis obtain through the Proper
Orthogonal Decomposition (POD) method. For that purpose, section 2presents each basis and their combination with
the PGD method. Each combination is compared both on its accuracy and computation time. The global methodology
applied is explained in section 3. To evaluate the basis in several situations, two case studies are presented. The first case
study, presented in section 4is a theoretical application. In section 5, the models are then applied to a practical case
with realistic boundary conditions. The results of the models will be confronted with laboratory measurements. For each
case study, the influence of several parameters on the accuracy of the approximation basis is investigated: the number of
modes in the approximation basis, the number of modes in the PGD model and the discretization.
2 Materials and Methods
2.1 Physical problem of heat transfer in building wall
The physical problem studied is defined to be as close as possible to problems usually solved by building energy models
(e.g. EnergyPlus [8]). It involves transient one-dimensional heat conduction through a wall without volumetric heat
dissipation for a time interval τwith t 0, τ and space interval xwith x 0,L:
cu(x,t)
t=
xku(x,t)
x,(1)
On each side of the wall, a Fourier boundary condition is assumed. On x = 0, the boundary condition can be
described by the following equation:
ku(x,t)
x= q(t) hout u(x,t) uout (t) ,x = 0 ,(2)
The surface energy balance depends on a net radiative heat flux, noted q, and a sensible heat flux. The last one is
calculated from the outdoor air temperature uout varying over time and from a convective heat transfer coefficient h out .
On x = L, the boundary condition can be described by the following equation:
ku(x,t)
x=hin u(x,t) uin(t) ,x = L .(3)
As it is illustrated with the outside boundary condition, another radiative heat flux could have been added to the inside
boundary condition to complexify the mathematical model proposed here. The net radiative heat flux is neglected on that
side of the wall. To support this hypothesis, the error due to this simplification of the mathematical model is studied in the
2/32
Parametric PGD model used with orthogonal polynomials to assess efficiently the building’s envelope thermal performance
appendix A. Note that this assumption will not have an impact on the results presented because the same mathematical
model is used for all the numerical models developed.
The sensible heat flux is calculated from the indoor air temperature uin that varies over time and from a convective
heat transfer coefficient h in .
The initial temperature is uniform:
u(x,t) = u 0,t = 0 .(4)
Equation (1) can be written in a dimensionless form as:
∂u(x, t)
∂t =F o 2u(x, t)
∂x 2,(5)
for a time interval Γ=0,Γand space interval x=0,1, and the boundary condition as:
∂u(x, t)
∂x =Biout uuout q , x = 0 ,(6a)
∂u(x, t)
∂x =Biin uuin , x = 1 .(6b)
The initial condition becomes:
u= 0, t = 0 .(7)
Where the dimensionless quantities are defined as:
u: = uu0
u0
;t: = t
tref
;x=x
L;Bi in : = hin.L
k;Bi out : = hout.L
k;F o : = k.tref
c L2= 1
tref : = c L2
k;uin : = 1 + uin
u0
;uout : = 1 + uout
u0
;q: = q.L
k.u0
; Γ = τ
tref
2.2 The related boundary value problem in the context of co-simulation
The physical problem involves the partial differential equation (PDE) Eq. (5) together with the boundary (Eqs. (6a)
and (6b)) and initial conditions (Eq. (7)). As presented on Figure 1, it is solved in the context of co-simulation (or
coupling) with other numerical models (models 1 and 2), by solving the radiative heat balance and the air transfer around
the building walls.
In this context, the initial boundary value problem Eq. (5) is semi-discretized along the time line [9]. The time
discretization parameter is denoted by ∆t, corresponding to the time step of coupling between the numerical models of
the co-simulation. The discrete values of functions u(x , t ) is written as undef
:= u(x , t n) with n= 1 ,... ,Nt. Thus,
using an implicit approach, Eq. (5) becomes:
un+1 =un+ ∆t·F o ·2un+1
∂x 2,(8)
By introducing yun+1 , Eq. (8) can be reformulated as:
ya·2y
∂x 2=b(x).(9)
Here, yis the unknown of our boundary value problem and depends on the space coordinate x. The coefficient adef
:= ∆t·F o
depends on the properties of the material composing the wall and on the co-simulation time step. The coefficient bdef
:= un
3/32
Parametric PGD model used with orthogonal polynomials to assess efficiently the building’s envelope thermal performance
Wall Model
Model 2
Wall Model
Model 2
Model 1 Model 1
Figure 1. Co-simulation process
is qualified as the source term of the boundary value problem, depending on the space coordinate x. It also varies at each
time step of the co-simulation. The boundary conditions Eqs. (6a) and (6b) are also transformed:
∂y
∂x =Biout ·ybout , x = 0 ,(10a)
∂y
∂x =Biin ·y+bin , x = 1 ,(10b)
where the coefficients bout and bin are:
bout =Biout ·uout (tn)q(tn), b in =Biin ·uin (tn).
Both are constants given at each time step ∆tof the co-simulation by model 1 and 2.
2.3 Formulation of the parametric problem
The boundary value problem Eq. (9) together with the boundary conditions (10) are the main interest to build a
reduced-order model. Several solvers exist to solve such a problem. A brief overview can be consulted in [10]. These
numerical models are used to compute a solution y(x) only depending on the space coordinate.
However, it is a challenging problem to build a solution depending on the space coordinate and on extra-parameters
such as the source term band the coefficients bout and bin . It requires to solve a so-called parametric problem. The use
of the PGD methods gives the opportunity to decompose the solution of a problem as a function of any parameters to
generate a parametric model.
Taking into account the source term as a parameter is another challenging task. Indeed, once discretized in space, the
source term is made of discrete values: one information per piece of the mesh. It implies inputting as many parameters in
the parametric model as the number of pieces of the mesh. To avoid this large number of involved parameters, the source
term is approximated by its projection on an approximation basis with a lower rank:
bx=
N
X
j= 1
Ψjxζj(11)
where Ψ jis the approximation basis, ζjare the coefficients of the projection and Nthe number of modes in the basis.
So, the solution of Eq. (9) is searched as:
u:h0,1i×bout ×bin ×ζjR,
x , b out , b in , ζ j7−ux , b out , b in , ζ j.
4/32
Parametric PGD model used with orthogonal polynomials to assess efficiently the building’s envelope thermal performance
The sets bout , bin and ζjare the domain of variations of the coordinates bout ,bin and ζj, respectively. They are
defined such as:
bout =b
out , b +
out ,bin =b
in , b +
in ,ζj=ζ
j, ζ +
j, j 0,N.
Their respective discretization parameters are denoted by ∆ bout , bin and ζj.
2.4 Approximation basis
In the literature, several parameterizations have been studied. Chinesta et al. (2013 [11]) and Gonzalez et al. (2012
[12]) proposed to use the nodal values corresponding to the piece-wise linear finite element approximation of the problem.
However, according to Gonzalez et al. (2014 [13]), this method leads to a large number of degrees of freedom: their
model is made of one parameter per nodal values. That is why they proposed to use the POD to provide a suitable
parameterization of the initial condition with the lowest number of degrees of freedom [13]. More information can be
found on this method applied to convective heat transfer in [14] and solid dynamics in [13,15].
One of the main drawbacks of this method is that a learning process is needed. It has an impact on the accuracy of
the reduced-order basis. For this reason, the data-set used must be representative of the problem (boundary values, initial
conditions, materials used).
According to Gonzalez et al. (2014 [13]) the initial condition could be interpolated by piece-wise polynomials. However,
for the specific field of solid dynamics, this approach is not the best choice, considering the behavior of the system. Another
solution proposed by Poulhaon et al. (2012, [16]) is to use an auxiliary mesh much coarser than the one used for the
solution of the problem. A projection is made from the fine to the coarse mesh using the least square method. This
method is purely mathematical and does not take into account physical considerations such as energy conservation or heat
flux conservation. According to Poulhaon et al. (2012, [16]) it should be completed by a mathematical tool to take into
account the physics of the studied phenomenon.
Conforming to [17], one important feature for the choice of an approximation basis is the sparsity. It ensures that the
chosen basis has the required regularity to represent the solution. Spectral basis, such as polynomial or trigonometric
functions, guarantee sparsity. For such functions, the values of the coefficients decrease exponentially with the order of
approximation [18]. However, the basis is full (because a spectral basis is not interpolative [17]). It implies that the
computational cost needed to determine the coefficients becomes impractical for large systems.
Based on the literature review, two methods are here compared: the use of a polynomials basis and the use of a POD
basis to approximate the temperature profile. Details on how to build each approximation basis used are given in the
appendix B.
As stated by the Weierstrass approximation theorem, every continuous function on a bounded interval can be
approximated by a polynomial to a certain accuracy [5]. Several functions with polynomial basis can then be used
according to the studied problem. The most simple polynomial basis is the monomial one. As described by Peyret
(2013 [19]), if a periodic problem is studied, the Fourier method should be used. Yet, this method is not suitable for
non-periodic problems, because of the Gibbs phenomenon. In this case, orthogonal polynomials such as Chebyshev or
Legendre polynomials should be used.
Considering the numerous polynomial basis, the first difficulty is to select the right basis for the considered problem.
As the field of interest is a non-periodic, smooth function, Fourier or Laurent polynomials shall not be studied here.
According to Trefethen (2013, [5]), the monomial basis is comfortable but should never be used to approximate a
function. If we compare the condition number for inversion of the three basis, the Chebyshev and Legendre polynomials
basis have a smaller condition number than the monomial one. If the condition number of a matrix is large, the matrix
is close to being singular. The condition number reveals that the projection of the field of interest on the monomial basis
will be sensitive to numerical round-off errors and perturbations in the input data. Moreover, monomial basis do not meet
sparsity condition as its coefficients increase with the order. Therefore, this basis should not be used here to parameterize
the initial condition.
5/32
Parametric PGD model used with orthogonal polynomials to assess efficiently the building’s envelope thermal performance
According to Trefethen (2013, [5]), Legendre points and polynomials are neither better than Chebyshev ones for
approximating functions, nor worse. The main advantage to use Chebyshev over Legendre points center around the
use of FFT (Fast Fourier Transform). This function can be used to get the coefficients from the point values or the
reverse. But this property is not used here. Both polynomials basis will be compared.
The Chebyshev and Legendre polynomials are part of the family of orthogonal polynomials. They are calculated
respectively at the Chebyshev and Legendre points. Special attention must be given to the spatial domain of the
problem. The points define a non-uniform mesh for a space interval [1,1]. Thus, a change of variable must be performed
to transform the dimensionless spatial domain [0,1] to x[1,1].
2.5 Proper Generalized Decomposition method
Several MOR methods can be used to solve a parametric problem. One of them is the Proper Generalized Decomposition
Method (PGD). It is an a priori MOR method based on the separation of variables. It does not reduce the system of
equations itself but the whole parametric problem. Any variable can then be defined as an extra-parameter of the model
[17].
With spectral methods [20], the PGD method is one of the unique methods that allows to create a complete parametric
model without knowing a priori the solution of the problem.
The PGD is used to propose an accurate parametric solution of the formulated BVP problem. The method approximates
the solution as a finite sum of separable functions. As presented in Section 2.3, the parametric model involves three
parameters: the space, the boundary condition and the source term. Applying the PGD method, the solution is sought
as the sum of Mfunctional products involving each function as follows:
y=
M
X
m= 1
XmxEmbin Fmbout N
Y
j= 1
Gj
mζj(12)
where X,E,F, and Gdesignate the functions of the parameters. Each function is defined over a domain : x=1,1,
bin =b
in, b+
in , Ωbout =b
out, b+
out and Ωζj=ζ
j, ζ+
j].
The following weak form of the ODE is used with the test function y(Galerkin formulation):
Zx×bin ×bout ×ζj
y.
ya2y
∂x 2
N
X
j= 1
Ψjxζj
dx. dbin.dbout .dζj= 0 (13)
The weak form of the ODE is regarded as an optimization problem. It leads to a nonlinear optimization problem due to
the functional product of the subspaces. It can be solved with an iterative procedure that features two nested loops: the
alternating direction strategy and the enrichment process [21]. The calculation of the unknowns is performed alternatively
along each dimension until convergence [22]. In this way, the algorithm splits the high dimensional problem into a series of
low dimensional ones. The complexity of the problem then grows linearly with the number of parameters. Each function
Xm,Em,Fmand Gj
mis first randomly initialized and then solved by iterations. The alternating directions process
stops once a fixed point is reached. The criterion ˜ǫused to make this determination is defined by the user [11]. Once
this criterion is reached, the new functions are added to the previous one in the PGD basis. The enrichment process
of the PGD basis stops when the ǫcriterion, defined by the user, is reached [11]. Details on the alternating directions
strategy equations and algorithms for a similar problem can be found on [3]. For further details on the method and its
developments, the interested reader may refer to [11,15].
Each function (Xm,Em,Fm,Gj
m) defined previously depends on a continuous variable. To solve the parametric
problem with the previous algorithm, the continuous variables need to be discretized. For that purpose, the continuous
variable is projected on a mesh. The continuous variable is then described by a vector. The finer the mesh of discretization
of each parameter, the closer the discrete value to the continuous one. But as a results, the number of elements in the
vectors used to describe the parameter increases.
6/32
Parametric PGD model used with orthogonal polynomials to assess efficiently the building’s envelope thermal performance
According to Leon et al. (2018, [23]), the final accuracy of a PGD model depends on the number of terms Min the
final sum, on the number of parameters/vectors ( x,bin ,bout,ζj) and the discretization of those parameters. However,
by increasing the number of elements in the mesh of discretization for each parameter, we increase the complexity of
the problem. In the case of a PGD model, this complexity grows linearly with the number of parameters [17,22]. As a
comparison, the complexity of a grid-based discretization (finite element, finite difference) grows exponentially with the
number of mesh elements. The number of elements on each vector is a matter of CPU time and space to save the PGD
parametric model. As the purpose of building a PGD parametric model is to decrease the calculation time (compared to
a classical model: finite difference, finite element) the number of elements in each vector should be then optimized.
For the spatial parameter and the boundary condition, the methodology to defined the discretization is classical, no
special interrogation arises. However, each mode of the approximation basis also needs to be discretized. Several questions
can arise for the coefficients ζjof the source term approximation. Spectral basis such as Chebyshev or Legendre
guarantee sparsity. When this condition is met, the order of magnitude of the coefficients ζjdecreases exponentially with
the order of approximation [18]. The discretization of each coefficient needs to fit the order of magnitude of each mode.
To simplify our study and only use one parameter to define the discretization of each coefficient of the basis, we propose
to use dimensionless numbers for the coefficients ζjdefined as ζj.
ζj=ζjmin (ζj)
max (ζj)min (ζj)(14)
where ζj[0,1] and ζj[min (ζj), max (ζj)].
2.6 Offline/online strategy
The use of the PGD method, to solve the parametric problem, features an offline-online strategy. During the offline
stage, the model is built for the set of parameters. It is then used online combined with other models. Online, the use of
the model requires no more than reading the unknown value in an abacus.
As previously described, one of the parameters of the problem consists of the source term b. Taking into account the
source term as a parameter is a challenging task. Indeed, once discretized in space, the source term is made of discrete
values: one information per piece of the mesh. It implies inputting as many parameters in the PGD parametric model as
the number of pieces of the mesh, plus the boundary conditions and spatial coordinates. The PGD method has shown
success for problems up to dimension 100. However, the efficiency of a parametric model depends on the number of
involved parameters [13,16].
To avoid this large number of involved parameters, one can gather some of them. Considering the source term, this is
usually done by using an approximation basis. The temperature field is projected on an approximation basis of a smaller
size. The use of different approximation basis is investigated in this work: Chebyshev,Legendre polynomials and the
POD reduced basis.
The PGD method is combined with the approximation basis to build a PGD parametric model for the previous
presented physical problem. Each step of the offline/online strategy is described in Figure 2.
The first step of the offline phase consists of building an approximation basis. In the case of Chebyshev,Legendre
basis, it is made of the polynomials. In the case of the POD reduced basis, a learning process is required. The POD
basis is built on a data-set. The latter can be provided from available measurements or from another model defined as
a Large Original Model (LOM). To get an accurate approximation basis, the learning process needs to be representative
of all future modeled combinations. In the specific case of a building energy model, the basis should be representative of
every material and climate data that could be used. The learning process needs a large amount of data and could be very
time-consuming.
The approximation basis Ψ aims at representing the source term in a minimum number of parameters called modes.
For that purpose, the approximation basis is truncated. A number of modes in the approximation basis, N, is defined to
achieve the desired approximation accuracy. Note that this number has a direct influence on the number of parameters
used in the PGD parametric model and its accuracy.
7/32
Parametric PGD model used with orthogonal polynomials to assess efficiently the building’s envelope thermal performance
Figure 2. Offline/online strategy
Then, all the parameters of the model (the mesh, the boundary conditions, and the approximation basis modes) are
converted into parameter vectors. The discretization (∆x, bout ,bin,ζ) selected for each vector has an impact on
the accuracy of the PGD parametric model.
Finally, as a last step of the offline phase, the parametric problem can be solved with the PGD algorithm. The PGD
parametric model is built for a number of PGD modes M. This parameter also influences the accuracy of the parametric
model.
Once the PGD parametric model has been built, it can be applied for any value within the previously defined intervals,
online. The source term bis projected on the approximation basis Ψ to identify the parameters ζj. Afterwards, the PGD
modes are computed for the defined parameters x,bin ,bout and ζj. The evaluation of the solution demands no more than
reading a look-up table [17].
3 Methodology
The purpose of this article is to overcome the obstacle of parameterizing the initial condition of a PGD parametric
model. It is then necessary to quantify and compare the accuracy of each approximation basis in the framework of their
combination with the PGD. The proposed study will therefore cover several issues:
1. the accuracy of the approximation basis for a given number of modes N,
2. the discretization of each of the parameters vectors,
3. the number of PGD modes M.
For the use of the POD approximation basis, a supplementary issue has to be added: the efficiency of the learning
process.
8/32
Parametric PGD model used with orthogonal polynomials to assess efficiently the building’s envelope thermal performance
3.1 Methods assessment’s procedure
To evaluate the approximation basis in several situations, two case studies are presented. The first case study is a
theoretical application. It is used to study the influence of the three first issues cited previously.
The built basis are then applied to a practical case with realistic boundary conditions. The results of the models will
be confronted with laboratory measurements. The influence of the learning period is studied through this second case
study.
They may seem simple and we could have considered more complicated case studies. However, the parametric model
would have been more complex. It would then have been more complicated to identify the influence of the studied
parameters on the final error of the model.
For each case study, the global methodology consists of two main steps. First, the approximation of the source term
is evaluated to study the behavior of the basis alone. Then the PGD parametric model is evaluated to verify if the basis
have the same behavior once applied in the PGD framework. The performance of the three basis is compared with regards
to the model errors and CPU time. The chosen indicators are presented hereafter.
3.2 Error indicator of the model
For each step of the assessment procedure, the error indicator chosen is the norm. It is computed as the Root
Mean Square Error between two spatial profiles. Only the maximum of the previous function is observed. This Section
describes the errors calculated for each of the three parameters studied in this paper.
3.2.1 Evaluation of the source term approximation
Source terms
Source term approximation
Figure 3. Evaluation of the source term approximation
For each metrics introduced hereafter, Figure 3summarizes the methodology. First, the performance of each basis to
approximate the source terms is evaluated by projecting the source terms (actual band reference bref ) on the different
basis and by then calculating the errors µas follows:
µ: ( N,Ψ) 7→ max
t
v
u
u
u
t1
Nx
Nx
X
0
bref
N
X
j= 1
Ψjζj
2
(15)
where Nxis the number of elements over the axis. The reference source term (noted bref ) outcomes from the reference
solution calculation at each point of the spatial mesh and for each studied time step. The error is calculated for each
approximation basis Ψ. The influence of the parameter Non the accuracy of the basis µwill be studied.
9/32
Parametric PGD model used with orthogonal polynomials to assess efficiently the building’s envelope thermal performance
To integrate the approximation basis into the PGD framework, each parameter of the model has to be discretized. The
error due to this discretization noted νis evaluated for each approximation mode. The error is calculated as follows for
the PGD variable ζ:
ν: ( N,Ψ, ζ, ζ)7→ max
t
v
u
u
u
t1
Nx
Nx
X
0
bref
N
X
j= 1
Ψjζj
2
(16)
with ζthe dimensionless coefficients.
3.2.2 Evaluation of the PGD parametric model
Figure 4. Methods assessment’s procedure sum-up
Finally, the approximation basis are introduced into a PGD parametric model to get a combined parametric model. The
error of the combined model noted εis computed between the calculated temperature profile and the reference solution.
Figure 4summarizes the methodology.
ε:N, ζ, Ψ,ζ , M 7→ max
t
v
u
u
u
t1
Nx
Nx
X
0
yref
M
X
m= 1
XmxEmbin Fmbout N
Y
j= 1
Gj
mζj
2
(17)
3.3 Indicator for the CPU time
A fair comparison of the computational time for various methods is not easy to undertake as it will depends on the way
we code and the tools we use. To get a fair comparison, calculation times were measured on the same computer and on
the same environment. Except for the reference solution calculation (where the Matlab toolbox Chebfun has been used),
we developed all the other computational codes by ourselves. We paid attention to code each model in the same way (for
example the same algorithm is always used to solve a system of equation).
For each step, the CPU calculation time is evaluated on a Lenovo, windows 10 with 8Go RAM IntelCore i5, 2.60 GHz.
The CPU calculation time is normalized by the time constant t0. It corresponds to the maximum CPU time observed.
This information will be given in the titles of the figures. The CPU time ratio is noted ρCPU and defined as follows:
ρCPU =tCP U
t0
(18)
10 /32
Parametric PGD model used with orthogonal polynomials to assess efficiently the building’s envelope thermal performance
4 Theoretical case study
4.1 Description of the case study
4.1.1 Physical constants used
The case study consists of a wall of one-layer of thickness L = 0.20 m, made of concrete, with a thermal conductivity
k = 1.75 W.m1.K1and a specific heat capacity c = 2.2 10 6J.m3.K1.
On the outdoor side of the wall, a sinusoidal variation of the air temperature and the net radiative heat flux are
considered. Their variations are defined as:
uout = u o,m+δo,1sin( 2 π ω o,1t) + δo,2sin( 2 π ω o,2t) (19)
q = q msin( 2 π ω q,t) 20 (20)
On the indoor side of the wall, a sinusoidal variation of the air temperature is considered, as described below:
uin = u i,m+δisin( 2 π ω i,1t) (21)
As presented before, the net radiative heat flux is neglected on that side of the wall. The error due to this simplification
of the mathematical model is studied for the specific case study in the appendix A.
The following numerical values are considered for the outdoor and indoor boundary conditions:
uo,m= 20 [ C], δ o,1=4.4 [ K], ω o,1=1
72 [h1], δ o, 2=11.7 [ K], ω o, 2=1
24 [h1],
qm= 500 [ W.m2], ω q, =1
48 [h1],ui,m= 20 [ C], δ i=2.0 [ K], ω i,1=1
48 [h1].
Some of the numerical values are inspired from 1D numerical application [2]. The boundary conditions used are presented
in the Figure 5. The convective heat transfer coefficients are set to : hin = 8.7W.m2.K1and h out = 23.2W.m2.K1.
0 10 20 30 40 50 60 70
h
5
10
15
20
25
30
35
C
(a) Inside and outside air temperature signal
0 10 20 30 40 50 60 70
h
0
100
200
300
400
500
W m
(b) Net radiative heat flux
Figure 5. Boundary conditions of the theoretical case study
The numerical values of the dimensionless quantities are the following ones:
Bi in : = 0.4971; Bi out : = 1.3314; F o : = 1; t ref : = 1.2571 ×104
11 /32
Parametric PGD model used with orthogonal polynomials to assess efficiently the building’s envelope thermal performance
0 10 20 30 40 50 60 70
h
10
15
20
25
30
35
C
(a) Indoor and outdoor surface temperature time series
0 0.05 0.1 0.15 0.2
m
14
16
18
20
22
24
26
28
C
(b) Temperature profiles inside the wall
Figure 6. Temperature field for the reference solution
4.1.2 Reference solution
The reference solution yref (x, t) is computed using the Matlab toolbox Chebfun [24] for a time horizon of 3 days, with
a dimensionless time step of ∆t= 103and a space mesh made of 200 nodes. The evolution of the temperature for the
reference solution is presented in Figure 6. Figure 6(a) describes the temporal evolution of the surface temperature on
each side of the wall and figure 6(b) gives an overview of the temperature profiles within the wall. It represents the source
term that needs to be parameterized with the several studied approximation basis.
4.1.3 Learning process
As presented in the Section 2.6, the POD basis is built on an available data-set. The choice was made to use the
complete reference solution data-set to built the POD basis. The POD basis is then used in identical conditions than the
one used for the learning process. Thus, the condition of the learning process will not influence the accuracy of the basis.
4.2 Evaluation of the approximation of the source term
The ability of each basis to approximate the source term depends on two parameters : the number of modes in the
approximation basis Nand the discretization of the parameters. The influence of those two criteria is studied hereafter.
4.2.1 Influence of the number of modes in the approximation basis
Figure 7presents the evolution of the approximation error as a function of the number of modes Nin the three
approximation basis. In Figure 7(a), we can observe that the error decreases as the number of modes in the approximation
basis increases. In the case of the POD basis, the error decreases until it gets constant around N= 18. The results of the
Chebyshev and the Legendre polynomial basis are very close. They both decrease with a large slope for the first ten
modes and continue to decrease slowly. The polynomial basis cross the POD basis around N= 38 modes. The polynomial
basis are then more accurate than the POD one.
The smoothness of the function can be linked to the number of times the function is differentiable. As explained by
Trefethen (2013 [5]), the smoother a function, the faster its approximates converge. Figure 7(b) gives information on the
smoothness of the function. In this specific case, the three approximation basis have similar trends. They converge at a
rate of O(N7).
12 /32
Parametric PGD model used with orthogonal polynomials to assess efficiently the building’s envelope thermal performance
0 10 20 30 40 50
10-10
10-8
10-6
10-4
10-2
(a) Semi-logarithmic scale
101
10-10
10-8
10-6
10-4
10-2
(b) Logarithmic scale
Figure 7. Evolution of the error µas a function of the number of modes in the approximation basis
10 20 30 40 50
0
0.2
0.4
0.6
0.8
1
CPU
Figure 8. Evolution of the CPU calculation time as a function of the number of modes in the three approximation basis
with t0= 0.2341 sec
13 /32
Parametric PGD model used with orthogonal polynomials to assess efficiently the building’s envelope thermal performance
The CPU calculation time is another criterion to compare the performance of the three basis. CPU time ratios are
presented on Figure 8. The results are normalized by the maximum CPU time observed (for Legendre basis with
N= 50). The calculation time presented for the POD basis includes the learning process. We can observe that the CPU
time increases linearly and that it is slightly higher for the POD basis than for the two polynomial basis. However, the
results are of the same order of magnitude.
4.2.2 Influence of the discretization
As reported in the Section 2.5, each parameter of the model (the mesh, the boundary conditions, and the coefficients
of the approximation basis) needs to be converted into vectors of parameters. For that purpose, their domain needs to
be discretized, by converting the continuous functions into discrete values. The mesh of discretization of the parameter ζ
has a direct impact on the accuracy of the approximation of the source term. The influence of the mesh of discretization
ζis studied hereafter.
Three dimensionless discretizations have been selected ∆ζ= 102,104,106. For the three criteria, the error
between the reference solution and the solution projected on the truncated basis is plotted as a function of the number of
modes in the truncated basis. Figure 9presents the results. For each curve, the same tendencies can be observed, the error
drops and then stabilizes. Indeed as we increase the number of modes, the error of the approximation decreases. However,
as the coefficients are rounded, part of the information is lost. As the error stabilizes, the addition of a supplementary
mode does not improve the accuracy of the approximation. For a discretization ∆ζ= 102, the threshold is reached for
N= 5 and 7 and 12 modes for respectively: ζ= 104and ∆ζ= 106.
4.2.3 Discussion
From those two first influence analyses, the approximation basis can not be ranked, as their performances are close.
For both the number of modes Nand the discretization, the same tendencies can be observed for the three approximation
basis.
Moreover, the accuracy of the POD basis depends on the quality of the learning process (it should be representative of
the conditions of future study cases). In the theoretical case study, the learning process has been made on the complete
reference solution data-set. We are then in ideal conditions for the use of the POD basis. In the practical application
(Section 5), the influence of the learning process will be investigated.
This first step enables the comparison of the behavior of the three studied basis outside of the PGD framework.
However, once implemented in the PGD framework, the tendencies observed before could be different. to verify the
consistency, the influence of the parameters studied should be studied in the PGD framework.
4.3 Evaluation of the PGD parametric model
A PGD parametric model is built to solve the problem studied here. The boundary conditions and the source term
are defined as parameters of the parametric model. The approximation basis are used to describe the initial condition in
a few parameters (modes). The PGD model is then combined to an approximation basis. The accuracy of the combined
model depends on three parameters:
the accuracy of the approximation basis for a given number of modes N,
the discretization of each of the parameters vectors,
the number of PGD modes M.
To study the influence of those three parameters on the accuracy of the model, several PGD basis have been generated,
one for each: combination of the three approximation basis, number of modes in the basis N [2,5] and discretization
ζ[105,102]. In total 48 PGD combined models have been compared. For each model, both parameters of the
alternating direction process and the enrichment process are fixed to ˜ǫ= 106and ǫ= 108. The influence of each
parameter ∆ζ, N, and Mis studied hereafter, based on the results of those basis.
14 /32
Parametric PGD model used with orthogonal polynomials to assess efficiently the building’s envelope thermal performance
2 3 4 5 6 7
10-4
10-3
10-2
(a) Evolution of the error for ∆ζ= 102
2 4 6 8 10
10-6
10-5
10-4
10-3
10-2
(b) Evolution of the error for ∆ζ= 104
2468 10 12 14
10-8
10-7
10-6
10-5
10-4
10-3
10-2
(c) Evolution of the error for ∆ζ= 106
Figure 9. Influence of the truncation of the approximation basis on the accuracy of the approximation for various
discretizations
15 /32
Parametric PGD model used with orthogonal polynomials to assess efficiently the building’s envelope thermal performance
(a) εerror for N= 5 (b) Online CPU calculation time ratio for N= 5
Figure 10. Influence of the discretization of the approximation coefficients on the error and on the CPU time ratio with
t0= 23879 sec.
4.3.1 Influence of the discretization of the approximation coefficient
The influence of the approximation coefficients is studied here. Figure 10 presents the results of the error and about
the calculation time. The results are displayed for the most accurate basis used made of N= 5 approximation modes.
The accuracy of the model and the CPU time of each model increase, as the discretization gets finer. Those two results
are in accordance with the previous ones. The discretization induces a loss of information. The continuous function is
converted into discrete values as it is done for a spatial mesh for any numerical method. The finer the mesh, the closer the
discrete representation to the continuous function. However, as we increase the discretization, we increase the number of
elements in the vector. The online CPU time then increases. The same tendencies are observed for basis made of 3 and
4 modes. For basis made of 2 modes, the same tendencies are observed for the POD. However for the Chebyshev and
Legendre basis, the error remains high and constant as we decrease the discretization. For both polynomial basis, using
2 modes is not enough to approximate the source term accurately.
4.3.2 Influence of the number of modes in the approximation basis
The influence of the number of modes in the approximation basis is now studied. Figure 11(a) presents the evolution
of the εerror as a function of this parameter. Results are presented for a fixed discretization of ∆ζ= 105for each
approximation basis. In the case of the Chebyshev and Legendre combined parametric models, the error decreases
with the number of modes. This phenomenon can be observed for fine discretizations (∆ζ= 104or ∆ζ= 105). For
coarser discretizations, the error remains constant as we increase the number of modes. Adding a supplementary mode is
not useful if the discretization remains constant.
In the case of the POD combined parametric model, for a fixed discretization, adding a supplementary mode will not
decrease the error of the model. The discretization will only have an impact on the error of the model. Here, the model
is trained and used on the same data-set. The results may have been different if only part of the data-set has been used
to train the basis. This point will be illustrated in the practical application (Section 5).
For every model, a threshold around O(103) is reached after a few modes. The error of the final PGD model is then
not mainly due to the approximation of the source term but also to other parameters: the discretization of the boundary
condition on x = L fixed at 103, the discretization of the boundary condition on x = 0 fixed at 104, the spatial grid
fixed at 102.
16 /32
Parametric PGD model used with orthogonal polynomials to assess efficiently the building’s envelope thermal performance
(a) For ∆ζ= 105, evolution of the εerror (b) For ∆ζ= 105, evolution of the CPU calculation time ratio
Figure 11. Influence of the number of modes in the approximation basis on the εerror and CPU time ratio with
t0= 23879 sec
4.3.3 Influence of the number of modes in the PGD basis
Figure 12. Influence of the number of PGD modes Mon the εerror for δζ = 104and N= 4.
The last parameter studied is the influence of the number of PGD modes M. Figure 12 presents the evolution of the
error as a function of the number of PGD modes Mfor N= 4. The parametric models are built for ∆ζ= 104for
each approximation basis. This Figure gives information on how fast the PGD strategy converges. There are not many
differences between the three methods. Applied to non-symmetric differential operators, the PGD algorithm converges
slowly as its optimality is not guaranteed [11]. The PGD could contain more terms than strictly needed.
Each time a mode is added to the parametric model, a new variable is added to the problem. The computational
domain becomes of higher dimension, it must cover not only the physical and boundary conditions coordinates but also
the parametric domain [25]. Adding a parameter increases the complexity of the tensor subspace. In the case of the
PGD, this complexity grows linearly with the number of dimensions [17,22]. Figure 13 illustrates the impact of adding a
17 /32
Parametric PGD model used with orthogonal polynomials to assess efficiently the building’s envelope thermal performance
Figure 13. Evolution of the total number of PGD modes as a function of the number of modes of the approximation
basis. Each curve of each basis corresponds to a different discretization ζ[105,102]
new parameter to the PGD parametric model on the total number of PGD modes. As we increase the number of modes
in the approximation basis, we increase the number of parameters in the parametric model. As soon in Figure 14 the
convergence rate of the algorithm decreases. Thus, the number of necessary PGD modes increases to achieve the desired
accuracy (˜ǫ= 106and ǫ= 108) as we increase the number of parameters involved.
(a) For δ= 104, evolution of the εerror (b) For N= 3, evolution of the εerror
Figure 14. Evolution of the εerror as a function of the number of modes for the Chebyshev basis
4.3.4 Discussion
For the approximation coefficients discretization and the number of modes in the approximation basis, the same tenden-
cies are observed than the one observed for the approximation of the source term. The comparison of the approximation
basis on the approximation of the source term gives a good first overview of the behavior of the basis.
18 /32
Parametric PGD model used with orthogonal polynomials to assess efficiently the building’s envelope thermal performance
However, two modes are not sufficient to approximate the source term with Chebyshev and Legendre combined
parametric models. For the POD basis, the final accuracy of the PGD model is reached with two modes for a fixed
discretization.
Finally, Leon et al. (2018, [23]) have shown on the Poisson equation that the final accuracy of a PGD model depends
on the discretization of the parameters and the number of terms Min the final sum. Indeed the finer the discretization
of each parameter, the closer will be the discrete values to the continuous one. However, as they decrease the mesh, they
increase the convergence rate of the PGD algorithm and the necessary number of PGD modes in the model. The same
tendencies can be observed here in Figure 14(b). It presents the evolution of the εerror as a function of the number
of modes for the Chebyshev basis with N= 3. The error decreases and then reaches a threshold. Then adding a
supplementary PGD mode to the parametric model is not sufficient to decrease the error of the model. The discretization
should be decreased.
5 Practical application
In the previous parts, the POD basis, as most of the time, has shown its optimality. However, as mentioned above, the
performance of the POD basis depends on the quality of the learning process. It should be representative of the boundary
conditions applied to the case study.
In the theoretical case study, the learning process has been made on the complete reference solution data-set. The
POD basis is then used in identical conditions than the one used for the learning process. The influence of the learning
process has not been studied yet.
To obtain a POD basis, a training data-set is necessary. It can be obtained from measurements or from another
numerical model. Both methods are expensive since a large range of data is needed. To give an example, if we want to
use the parametric model to predict the temperature distribution in a wall during a year, the training data-set should be
representative of all the boundary conditions encountered in practice.
To illustrate this limit, the accuracy of various POD basis are compared to the polynomial basis. The same methodology
as the one used for the theoretical case study is applied. The influence of the learning period is first studied on the
approximation itself and then on the combination of the approximation basis with the PGD parametric model.
Another major objective of this part is to evaluate the reliability of the model in a realistic case study. For that
purpose, the results of the model are compared to laboratory measurements.
5.1 Description of the case study
5.1.1 Experimental set-up
The experimental set-up described hereafter was think up with the objective to obtain realistic boundary conditions and
measurements on a common building wall. It consists of a multi-layer building wall, made of traditional building materials:
1cm of plasterboard, 10 cm of insulation (expanded polystyrene), 15 cm of structural material and approximately 1 cm of
mineral coating. The wall is built between two rooms. One can be heated by an electric heater and the second one can
be cooled by the evaporator of a heat pump.
For this study, we will only focus on the insulation layer of the wall. Indeed, the insulation material experiences greater
temperature gradients which makes it more interesting to observe. Moreover, the insulation material is a homogeneous
material and the temperature is easier to measure in such a material, contrary to the structure material made of concrete
cellular blocks for which the measured temperature is strongly dependent on the position of the sensor. Indeed, for such
cellular materials, the measured temperature can be very different whether it is measured on a cavity or near the wall
of this cavity that creates a thermal bridge. By the more, the thermal conductivity of the insulation material is well
known, whereas only the macroscopic thermal resistance is known for the concrete cellular block, which makes it difficult
to obtain a calculated temperature directly comparable to the measured temperature, although the heat flux is correct.
This insulation layer is thus equipped with four type K thermocouples located at the surface and in the insulation layer.
19 /32
Parametric PGD model used with orthogonal polynomials to assess efficiently the building’s envelope thermal performance
Figure 15. Sensors position illustration and
nomenclature Figure 16. Evolution of the air temperature and inside
wall temperatures measurements
The global experimental uncertainty has been calculated with equation 22 [26].
σ=sσ2
m+∂u
∂x δx2
(22)
The thermocouples have been calibrated by measuring the temperature of melting ice and boiling water before the
measurement. The sensor measurement uncertainty is then σm=±0.1C. The sensor position uncertainty has been
evaluated as the product of the temperature derivative (with second-order centered approximation) at the sensor position
and δx =±0.1cm. The temporal mean global experimental uncertainty is noted hereafter σ.
5.1.2 Experimental observations
Data were recorded for 5 days, with a 30sec time step. Several cycles were tested during this period, turning on and off
the heater and/or the heat pump. The cycles are described in Table 1. A pattern made of three cycles with three different
time periods (25 min, 40 min, and 60 min) is repeated twice. The first three cycles are run with a temperature set-point of
5Cin the cold room. For the last three cycles, the heat pump was turned off to modify the boundary conditions of the
cold room. The boundary conditions are described through the evolution of the air temperature in the warm and cold
room in Figure 16.
5.1.3 Reference solution
The reference solution yref (x, t) of this problem is computed using a Euler implicit finite difference model for a time
horizon of 96 hours with a time step of 30sec (dimensionless time step of ∆ = 102) and a spacial mesh made of 99 nodes.
On each side, two Dirichlet boundary conditions are set using the temperature signal T01 and T04. Figure 16 gives an
overview of the temperature evolution at the boundary conditions.
The first temperature profile is initialized using the temperature profile measured in the wall at the beginning of the
experiment. Linear interpolation is done between the measured points to obtain the temperature distribution at each point
of the spatial mesh (Figure 17). The simulation is then run for the all period (5 days). The first 87hof the simulation
are not used. They are left as initialization period of the model. It consists of turning on the heater and the heat pump
until an equilibrium between the two rooms is reached. The boundary conditions of this initialization cycle are described
in Table 1. The rest of the data-set is used to evaluate the model in different conditions. As the first 87hare not used
to evaluate the model, they are not presented in the following figures. Thermal properties from the French regulations
database [27] are used with: k= 0.04 W.m1.K1and c= 30.103J.m3.K1.
20 /32
Parametric PGD model used with orthogonal polynomials to assess efficiently the building’s envelope thermal performance
Cycle number Heater Heat Pump Duration [min] Time
Initialization on on (5 C) 5220 (87h) -
0 off on (5 C) 40 00:00 to 00:40
1 on on (5 C) 40 00:40 to 1:20
1 off on (5 C) 40 1:20 to 2:00
2 on on (5 C) 25 2:00 to 2:25
2 off on (5 C) 25 2:25 to 2:50
3 on on (5 C) 60 2:50 to 3:50
3 off on (5 C) 60 3:50 to 4:50
4 on off 40 4:50 to 5:30
4 off off 40 5:30 to 6:10
5 on off 25 6:10 to 6:35
5 off off 25 6:35 to 7:00
6 on off 60 7:00 to 8:00
6 off off 60 8:00 to 9:00
Table 1. Description of the cycles
Figure 17. Initialization temperature profile
21 /32
Parametric PGD model used with orthogonal polynomials to assess efficiently the building’s envelope thermal performance
5.1.4 Learning process
Three training data-sets for the POD basis are compared:
1. the full evaluation data-set (noted tτ= [0 , τ] with τ= 9h),
2. half of the evaluation data-set, made of the cycles 0 to 3 (noted tτ
2= [0 ,4h50]),
3. the cycle 1 (noted tt1= [0 ,0h40]).
The three basis are compared to the Chebyshev and Legendre polynomial basis. For those two last methods, no
learning period is required to build the basis.
5.2 Influence of the learning period
As for the previous case study, the influence of the learning period is first evaluated on the approximation of the source
term, then it is evaluated on the PGD parametric model. The same parameters and computation code than the one used
for the previous sections are applied.
5.2.1 Evaluation of the approximation of the source term
The accuracy of the approximation of the source term is studied for various training data-sets. The results are presented
on figure 18. The error is plotted for the various number of modes in the approximation basis: N [2,8].
Figure 18. Approximation basis error as a function of the number of modes for several training periods
As in the theoretical example, the POD basis is the most accurate one for N [2,8], if the full data-set is used for the
training period. However if only a part of the data is available, the Chebyshev and Legendre approximation basis are
more efficient for N [2,3]. The POD basis trained with half of the cycles seems to be as efficient as the one built with
the full training data-set for N [4,8]. Indeed the same pattern is repeated from cycle 1-2-3 to cycle 4-5-6. Building the
POD basis with one pattern could be enough for N>3.
This learning process has a numerical cost as it requires running a large original model and building the POD basis as
described in Section 2.6. Table 2compares the computation time needed to build the basis from the results of the finite
difference model for the various learning periods. As large is the training data-set as large is the time needed. Building
the basis with one cycle results in a saving of 35% of the offline computation cost.
Finally, a compromise should be found to minimize the training period and the computational cost needed to build
the basis while keeping an accurate approximation basis. For that purpose, a methodology to select an efficient training
period should be developed.
22 /32
Parametric PGD model used with orthogonal polynomials to assess efficiently the building’s envelope thermal performance
Learning period ρCP U
tτ1
tτ
20.89
tt10.65
Table 2. Offline calculation time t0= 0.003987 sec
5.2.2 Evaluation of the PGD parametric model
The influence of the learning period is now studied for the combination of the PGD parametric model with the various
approximation basis. Results for the most favourable (tτ) and unfavourable (tt1) POD basis are compared to
the Chebyshev and Legendre polynomial basis. Several PGD basis have been generated one for each: combination of
the four approximation basis (the favourable POD, the unfavourable POD, the Chebyshev and Legendre polynomial
basis), number of modes N [2,5] and discretization ∆ζ[105,104]. In total 32 PGD basis have been compared for
this application. As done before, both parameters of the alternating direction process and the enrichment process are
fixed to ˜ǫ= 106and ǫ= 108. The accuracy of the PGD parametric model is compared for various number of modes
N [2,5] and for a fixed discretization ∆ζ= 105. We use the same parameters than in section 4.3.2 to compare the
results.
(a) For ∆ζ= 105, evolution of the εerror (b) For ∆ζ= 105, evolution of the CPU time ratio
Figure 19. PGD parametric model εerror and CPU time ratio as a function of the number of modes for several training
periods ( t0= 51.31 sec)
Figure 19 presents the evolution of the error and CPU time for various numbers of modes N. For each combined
parametric model, for N [2,4] the error decreases with the number of modes. The error for N= 5 increases. This can
also be observed in Figure 11. As previously explained, a threshold (around O(103)) is reached after a few modes. This
phenomenon can be observed for both discretizations. One this threshold has been reached, the error of the final PGD
model is then not mainly due to the approximation of the source term. This could explain the fact that the error slightly
increases.
In the theoretical case study, the error of the POD basis remained constant with the number of modes. Adding
supplementary modes did not improve the total accuracy of the model. It is not the case here. For more complex boundary
conditions (realistic signal), supplementary modes are necessary to accurately parametrize the previous temperature profile.
23 /32
Parametric PGD model used with orthogonal polynomials to assess efficiently the building’s envelope thermal performance
For this practical example, the Chebyshev and Legendre polynomial basis are more accurate once combined with the
PGD basis for a similar computational time. This could be due, once more, to the complexity of the boundary condition
signal. It could be also due to the discretization ∆ζ. To encounter the same method ranking as the one presented in
Figure 18, the POD coefficients may need to be discretized more finely.
5.3 Comparison with experimental data
Finally, the ability of the PGD parametric model to reproduce the dynamics on a realistic example is here studied.
The results of the four models for N= 3 and ∆ζ= 105are compared to the measurements.
Figure 20 presents the time evolution at the position of sensors T02 and T03, respectively at 4 cm and 8 cm from
the inner surface. All four models follow the dynamics of the measured curve. In figure 20(a), we can observe that the
unfavorable POD basis matches the favorable POD basis for the first cycles, then the two curves depart from each other.
It denotes the fact that the POD basis will be accurate as it encounters its training boundary conditions but will deviate
as it encounters different boundary conditions.
Figure 21 presents the error to the measurement data at both depth 4 cm and 8 cm from the inner boundary condition
for the various numbers of modes N. The same tendencies are observed as the ones described for figure 19. The error
decreases and stabilizes after a few modes for each model. Depending on the reference data, 4 cm and 8 cm, the method
ranking is not the same. Results for 4 cm are similar to the one observed comparing the PGD solution to the reference
solution (finite difference model). In the results for 8 cm, we can see that the training period of the POD basis has less
influence. Indeed at this location, the signal amplitude is eased. It fluctuates less. It could be easier to parameterize this
part of the temperature profile.
This last study confirms the ability of the PGD parametric model with the approximation basis to reproduce the
dynamics of the signal. At 4 cm, for N>2, all four models reach the mean experimental uncertainty. At 8 cm, for N>2,
the Legendre combined models is getting closer to σ, while the other models errors are under the threshold of the mean
experimental uncertainty.
Finally, the accuracy of the models is of the same order of magnitude than the reference solution. Indeed, if we
quantify the error between the reference solution and the measurements, we obtain an error of 0.18 Cat 4 cm from
the left boundary condition and 0.13 Cat 8 cm from the inner boundary condition. Those values are close to the one
presented in Figure 21.
6 Conclusions
The POD, the Chebyshev and Legendre polynomial approximation basis have been compared first on a theoretical
example. This case study was an opportunity to quantify the influence of three main parameters:
1. the number of modes Nin the approximation basis,
2. the discretization coefficient,
3. the number of modes Min the PGD basis.
The different basis were then compared on a practical example based on measurements. This second case study intended
to highlight the influence of the learning process on the accuracy of the POD basis. It also enables the comparison of the
three combined PGD parametric models with measurements.
The approximation basis have been first applied to the approximation of the source term. This first step has shown
that the discretization should be selected in accordance with the number of modes N. Indeed, increasing the number of
modes with an insufficient discretization will not increase the accuracy of the approximation.
24 /32
Parametric PGD model used with orthogonal polynomials to assess efficiently the building’s envelope thermal performance
(a) Temperature at 4cm from the inner boundary condition
(b) Temperature at 8cm from the inner boundary condition
Figure 20. Time evolution of the temperature measured and calculated by the models at various depths. The grey zone
corresponds to ±σ, the global experimental uncertainty.
25 /32
Parametric PGD model used with orthogonal polynomials to assess efficiently the building’s envelope thermal performance
(a) For ∆ζ= 105, evolution of the εerror at 4cm from the inner
boundary condition
(b) For ∆ζ= 105, evolution of the εerror at 8cm from the inner
boundary condition
Figure 21. PGD parametric model εerror to the measurement data (σcorrespond to the mean experimental uncertainty)
The different approximation basis were then integrated into the PGD parametric model. The first study on the
influence of the discretization of the approximation coefficient revealed that the accuracy and the computation time are
proportional to the discretization. The finer the mesh, the closer the discrete representation to the continuous function.
However, as we increase the discretization, we increase the online calculation time.
The study on the influence of the number of modes Nhas shown that the error decreases as we increase the number
of modes in the approximation basis. This is not the case when the final accuracy is reached with a few modes as it was
the case for the POD basis in the theoretical part. Finally, as the number of modes is increased, the computational time
increases.
A relation has also been highlighted between the number of approximation modes Nand the convergence rate of
the fixed-point algorithm. As the number of modes increases, the number of parameters in the PGD model increases,
decreasing the convergence rate of the algorithm. More modes Mare then necessary for the PGD basis to achieve the
same accuracy.
The efficiency of the PGD parametric model depends on the three basis on the three previous parameters studied.
A compromise should be found between the number of modes Nand M, the discretization and the computation time
needed to compute and use the PGD combined model.
The POD approximation basis has the main drawback to require a learning process. The benefit from a PGD parametric
model as an a priori method is then canceled out by the use of an a posteriori method. The combined POD and PGD
parametric model becomes then an a posteriori model. Its performance depends on the training data-set used.
The influence of this last parameter has been studied in the practical study case. Depending on the data-set used to
train the POD basis, it could be the most or the less accurate method to parameterize the source term.
Finally, a compromise should be found to minimize the training period and the computational cost needed to build
the basis while keeping an accurate approximation basis. For that purpose, a methodology to select an efficient training
period should be developed. This is a point of current work. Some leads have been explored on how to improve the
necessary training period by Berger et al. (2018, [2]). A methodology has been proposed in [28] to select a short and
representative training period for a building wall.
26 /32
Parametric PGD model used with orthogonal polynomials to assess efficiently the building’s envelope thermal performance
As a final conclusion, we should keep in memory that the POD basis provides an optimum basis if the learning process
is complete (the full data-set is used to build the basis). An efficient training data-set is then needed. However, when
those data are not available, polynomial basis are a good alternative. They have the main benefit to provide an a priori
combined PGD parametric model.
However for both methods, the POD or the polynomial approximation, this work should be continued. For the POD
method, the learning process remains the main barrier. For polynomial approximation, the parameterization of multi-
material wall brings to light new questions. With a multi-layer wall, the source term may not be a smooth function. The
efficiency of the polynomial basis to parameterize the temperature profile should then be tested.
Finally, the PGD model combined with each basis as shown its abilities to represent a realistic case study. Those
models are ready to be aggregated with other sub-models through a co-simulation process to replace a large original
model.
A Details on the model error due to the inside radiative heat flux
As mentioned in Section 2.1, the net radiative heat flux have been neglected on the inside part of the wall. This heat
flux is composed of the short and long-wave radiative heat flux. The short-wave radiative heat flux transmitted through
the building windows is generally taken into account and distributed to the building interior surfaces (by solar tracking or
with a weighted method) [29]. For the long-wave radiative heat flux, it calculation requires the introduction of non-linear
terms, most building simulation tools proposed then simplifications. This heat flux is either neglected, either linearised,
and integrated into the convective heat transfer coefficient.
A.1 Model error for the hypothesis neglecting the inside radiation effects
To evaluate the impact of neglecting the inside net radiative heat flux, it is possible to propose a model error for this
hypothesis. To obtain this model, the solution of the heat transfer equation considering inside radiation effects is denoted
by eu. Then, the boundary condition on the inside part of the wall is:
keu
∂x =hin euuin +qin , x =L ,
where qin is the incident radiation flux arising from the boundary surfaces facing the studied wall. The error between the
solutions is defined by:
edef
:= ueu . (23)
Recalling that uis the solution of equation 1, which neglect the inside net radiative heat flux. Since the problem is linear,
the model error verifies the following governing equation:
c∂e
∂t =
∂x k e
∂x ,(24)
with the following boundary conditions:
k∂e
∂x =hout e , x = 0 ,(25a)
k∂e
∂x =hin eqin , x =L , (25b)
and the initial condition:
e= 0 , t = 0 .(26)
The model error equations (24)–(26) can be computed using any of the numerical method presented in Section 2.4 and
Section 2.5. This is facilitated by working with dimensionless equations enabling to reuse the same numerical model for
different problems.
27 /32
Parametric PGD model used with orthogonal polynomials to assess efficiently the building’s envelope thermal performance
A.2 Results for the theoretical case study
The use of the model error is illustrated for the case study defined in Section 4. The inside radiative heat flux is defined
through long-wave radiation exchanges with surrounding surfaces:
qin =fwǫwσ4T4u4
w+fgǫgσT4u4
g,
where σis the Boltzmann constant and ǫw/g the emissivity of the material. uwand ugare the surrounding walls and
ground surface temperatures, respectively. The corresponding shape factor are fwand fg. The first part of the formula
corresponds to the radiative balance with the three walls and the ceiling, while the second part corresponds to the balance
with the floor.
For the numerical applications, the following values are considered:
fw=fg= 0.2, ǫw=ǫg= 0.9, σ = 5.67 ·10 8W.m2.K4, u w=uin , u g= 23 C.
To obtain the previous numerical values, the followings hypothesis have been made: - the room studied has no windows, -
the room is perfectly cubic (all the shape factors are equal to 0.2), - the surface temperatures of the walls and ceiling are
equal to the air temperature (an equilibrium has been reached with neighboring rooms), - the floor surface temperature
equal to 23 C(underfloor heating).
The flux qin is computed using a posteriori results of the wall. The time variation of the flux is shown in Figure 22(a).
It can be remarked that the radiation flux scales between 25 and 30 W.m2. It has a very low magnitude compared
to the outside flux, illustrated in Figure 5(b). Using the time variation of qin , the model error is computed based on a
finite-difference model. The time variation of the model error is given in Figure 22(b). The error reaches a maximum of
1.0Clocated, as expected, on the inside boundary (x=L). The impact of the hypothesis neglecting the inside flux can
be evaluated on the temperature flux. For this, the solution euis reconstructed using Eq. (23). The temperature variation
are illustrated in Figures 23(a) and 23(b). On the outside surface, the two solutions are almost overlapped. Thus, the
influence of the inside radiation is negligible on this part. Indeed, as remarked in Figure 22(b), the model error scales
with 0.2C. On the inside surface, the discrepancy between the solution is higher, around 0.5C.
As a synthesis, a model error is proposed to evaluate the influence of the hypothesis neglecting the inside net radiative
heat flux. It can be computed using any of the numerical models proposed in the manuscript, due to the benefits of
working with dimensionless equations. In terms of physical results, the inside radiation effects induce discrepancies on
the inside surface of the wall. However, the overall dynamics of heat transfer is not altered. Note that the numerical
investigations carried in Sections 4can be straightforwardly extended to a model considering inside radiation flux.
B Details on the approximation basis construction
B.1 Chebyshev polynomials
The Chebyshev polynomials are part of the family of orthogonal polynomials. The first kind Chebyshev polynomial
denoted Tnare the following ones:
T0x= 1, T 1x=x, T 2=2x21, T 3=4x33x.(27)
They are constructed according to the following relation of recurrence [5,19]:
Tj+ 1 = 2 x T jTj1for j > 1 with T0= 1,and, T1=x(28)
The Chebyshev approximation basis is made of the Chebyshev polynomials.
ΨjTj(29)
The Chebyshev polynomials are calculated at the Chebyshev points defined by the equation (30), where nis a
positive integer. In the literature several names can be found to describe this set of points as Chebyshev–Lobatto
points, Chebyshev extreme points, or Chebyshev points of the second kind. All those expressions refer to the same set
of points according to Trefethen (2013, [5]).
28 /32
Parametric PGD model used with orthogonal polynomials to assess efficiently the building’s envelope thermal performance
0 12 24 36 48 60 72
-25
-20
-15
-10
-5
0
5
10
15
20
25
30
(a)
0 12 24 36 48 60 72
-1.8
-1.6
-1.4
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
(b)
Figure 22. Time evolution of the inside boundary flux due to long-wave radiation (a) and of the model error (b)
0 12 24 36 48 60 72
10
15
20
25
30
35
(a)
0 12 24 36 48 60 72
15
16.5
18
19.5
21
22.5
24
25.5
27
(b)
Figure 23. Time evolution of the temperature with or without neglecting the inside radiation flux on the outside (a) and
inside (b) surfaces.
xj= cos j π
n,0< j < n, (30)
Special attention must be given to the spatial domain of the problem. The Chebyshev points define a non-uniform
mesh for a space interval [1,1]. Thus, a change of variable must be performed to transform the dimensionless spatial
domain [0,1] to x[1,1].
29 /32
Parametric PGD model used with orthogonal polynomials to assess efficiently the building’s envelope thermal performance
B.2 Legendre polynomials
The Legendre polynomials are also part of the family of orthogonal polynomials. The first Legendre polynomials are
the following ones:
P0x= 1, P 1x=x, P 2=3
2x21
2.(31)
The next polynomials are constructed according to the following relation of recurrence [5]:
(j+ 1 ) Pj+ 1 = ( 2 j+ 1 ) x P jj P j1for j1,and P0x= 1 , P 1x=x(32)
The Legendre approximation basis is made of the Legendre polynomials calculated at the Legendre points.
ΨjPj(33)
As for Chebyshev, special attention must be given to the spatial domain. The spatial mesh will not be uniform and a
change of variable must be performed to transform the dimensionless spatial domain from [0,1] to x[1,1].
B.3 POD reduced basis
The POD method extracts the relevant information from a set of snapshots by means of its projection onto a smaller
subspace. As a result, from a data-set, the POD builds a deterministic representation, from the basis Φ. The ultimate
goal is to retain a detailed representation of the data-set with a minimum or optimal number of modes in Φ. For these
properties, the POD method could be used to parameterize the temperature profile (source term in our problem).
ΨjΦj(34)
To build the POD basis, a learning process is needed. It has an impact on the accuracy of the reduced-order basis. For
this reason, the data-set used must be representative of the problem (boundary values, initial conditions, materials used).
More details on the POD methods can be found in [30,31].
Contrary to the two previous basis, no special attention needs to be paid to the definition of the spatial domain. To
standardize the spatial domain used, the same change of variable is performed (x[1,1]) and the spatial mesh is set
uniform.
References
[1] Laurent Malys, Marjorie Musy, and Christian Inard. Microclimate and building energy consumption: Study of
different coupling methods. Advances in Building Energy Research, 9(2):151–174, 2015. 1
[2] Julien Berger, Walter Mazuroski, Ricardo CLF Oliveira, and Nathan Mendes. Intelligent co-simulation: neural
network vs. proper orthogonal decomposition applied to a 2d diffusive problem. Journal of Building Performance
Simulation, pages 1–20, 2018. 1,11,26
[3] Marie-Hélène Azam, Sihem Guernouti, Marjorie Musy, Julien Berger, Philippe Poullain, and Auline Rodler. A mixed
pod–pgd approach to parametric thermal impervious soil modeling: Application to canyon streets. Sustainable Cities
and Society, 42:444–461, 2018. 2,6
[4] Julien Berger and Nathan Mendes. An innovative method for the design of high energy performance building envelopes.
Applied Energy, 190:266–277, 2017. 2
[5] Lloyd N Trefethen. Approximation theory and approximation practice, volume 128. Siam, 2013. 2,5,6,12,28,30
[6] Suelen Gasparin, Julien Berger, Denys Dutykh, and Nathan Mendes. Solving nonlinear diffusive problems in buildings
by means of a spectral reduced-order model. Journal of Building Performance Simulation, pages 1–20, 2018. 2
[7] Suelen Gasparin, Denys Dutykh, and Nathan Mendes. A spectral method for solving heat and moisture transfer
through consolidated porous media. International Journal for Numerical Methods in Engineering.2
30 /32
Parametric PGD model used with orthogonal polynomials to assess efficiently the building’s envelope thermal performance
[8] Drury B Crawley, Linda K Lawrie, Frederick C Winkelmann, Walter F Buhl, Y Joe Huang, Curtis O Pedersen,
Richard K Strand, Richard J Liesen, Daniel E Fisher, Michael J Witte, et al. Energyplus: creating a new-generation
building energy simulation program. Energy and buildings, 33(4):319–331, 2001. 2
[9] Suelen Gasparin, Julien Berger, Denys Dutykh, and Nathan Mendes. An adaptive simulation of nonlinear heat and
moisture transfer as a boundary value problem. International Journal of Thermal Sciences, 133:120–139, 2018. 3
[10] Lawrence F Shampine, Jacek Kierzenka, and Mark W Reichelt. Solving boundary value problems for ordinary
differential equations in matlab with bvp4c. Tutorial notes, 2000:1–27, 2000. 4
[11] Francisco Chinesta, Roland Keunings, and Adrien Leygue. The proper generalized decomposition for advanced nu-
merical simulations: a primer. Springer Science & Business Media, 2013. 5,6,17
[12] David González, F Masson, F Poulhaon, Adrien Leygue, Elías Cueto, and Francisco Chinesta. Proper general-
ized decomposition based dynamic data driven inverse identification. Mathematics and Computers in Simulation,
82(9):1677–1695, 2012. 5
[13] David González, Elías Cueto, and Francisco Chinesta. Real-time direct integration of reduced solid dynamics equa-
tions. International Journal for Numerical Methods in Engineering, 99(9):633–653, 2014. 5,7
[14] Victor Zucatti, Hugo FS Lui, Diogo B Pitz, and William R Wolf. Assessment of reduced-order modeling strategies
for convective heat transfer. Numerical Heat Transfer, Part A: Applications, 77(7):702–729, 2020. 5
[15] Elías Cueto, David González, and Icíar Alfaro. Proper Generalized Decompositions. SpringerBriefs in Applied Sciences
and Technology. Springer International Publishing, Cham, 2016. DOI: 10.1007/978-3-319-29994-5. 5,6
[16] Fabien Poulhaon, Francisco Chinesta, and Adrien Leygue. A first step toward a pgd-based time parallelisation strategy.
European Journal of Computational Mechanics/Revue Européenne de Mécanique Numérique, 21(3-6):300–311, 2012.
5,7
[17] Domenico Borzacchiello, José V Aguado, and Francisco Chinesta. Non-intrusive sparse subspace learning for
parametrized problems. Archives of Computational Methods in Engineering, pages 1–24, 2017. 5,6,7,8,17
[18] John P Boyd. Chebyshev and Fourier spectral methods. Courier Corporation, 2001. 5,7
[19] Roger Peyret. Spectral methods for incompressible viscous flow, volume 148. Springer Science & Business Media,
2013. 5,28
[20] Suelen Gasparin, Julien Berger, Denys Dutykh, and Nathan Mendes. Advanced reduced-order models for moisture
diffusion in porous media. Transport in Porous Media, 124(3):965–994, 2018. 6
[21] Francisco Chinesta, Pierre Ladeveze, and Elías Cueto. A short review on model order reduction based on proper
generalized decomposition. Archives of Computational Methods in Engineering, 18(4):395, 2011. 6
[22] Amine Ammar, Francisco Chinesta, Pedro Diez, and Antonio Huerta. An error estimator for separated representations
of highly multidimensional models. Computer Methods in Applied Mechanics and Engineering, 199(25-28):1872–1880,
2010. 6,7,17
[23] Angel Leon, Anais Barasinski, Emmanuelle Abisset-Chavanne, Elias Cueto, and Francisco Chinesta. Wavelet-based
multiscale proper generalized decomposition. Comptes Rendus Mécanique, 346(7):485–500, 2018. 7,19
[24] Tobin A Driscoll, Nicholas Hale, and Lloyd N Trefethen. Chebfun guide, 2014. 12
[25] Etienne Pruliere, Francisco Chinesta, and Amine Ammar. On the deterministic solution of multidimensional paramet-
ric models using the proper generalized decomposition. Mathematics and Computers in Simulation, 81(4):791–810,
2010. 17
[26] John R Taylor. Error analysis. Univ. Science Books, Sausalito, California, 1997. 20
[27] Journal officiel de la République Fançaise. Th-b-ce, 2012, fascicule th-u matériaux, 2017.
https://www.rt- batiment.fr/batiments-neufs/reglementation-thermique-2012/textes-de-references.html.
20
31 /32
Parametric PGD model used with orthogonal polynomials to assess efficiently the building’s envelope thermal performance
[28] Marie-Hélène Azam, Sihem Guernouti, Marjorie Musy, and Philippe Poullain. How to perform an efficient learning
process for a combined pod and pgd soil urban thermal model? application to canyon streets. In Proceedings of the
16th IBPSA Conference Rome, Italy, volume 29, pages 3155–3162, Sept. 2-4, 2019. 26
[29] Nicolas Lauzet, Auline Rodler, Marjorie Musy, Marie-Hélène Azam, Sihem Guernouti, Dasaraden Mauree, and
Thibaut Colinart. How building energy models take the local climate into account in an urban context–a review.
Renewable and Sustainable Energy Reviews, 116:109390, 2019. 27
[30] YC Liang, HP Lee, SP Lim, WZ Lin, KH Lee, and CG Wu. Proper orthogonal decomposition and its applica-
tions—part i: Theory. Journal of Sound and vibration, 252(3):527–544, 2002. 30
[31] Elías Cueto, Francisco Chinesta, and Antonio Huerta. Model order reduction based on proper orthogonal decompo-
sition. In Separated Representations and PGD-Based Model Reduction, pages 1–26. Springer, 2014. 30
32 /32
ResearchGate has not been able to resolve any citations for this publication.
Article
Full-text available
Separated representations at the heart of Proper Generalized Decomposition are constructed incrementally by minimizing the problem residual. However, the modes involved in the resulting decomposition do not exhibit a clear multi-scale character. In order to recover a multi-scale description of the solution within a separated representation framework, we study the use of wavelets for approximating the functions involved in the separated representation of the solution. We will prove that such an approach allows separating the different scales as well as taking profit from its multi-resolution behavior for defining adaptive strategies.
Article
Full-text available
We discuss the use of hierarchical collocation to approximate the numerical solution of parametric models. With respect to traditional projection-based reduced order modeling, the use of a collocation enables non-intrusive approach based on sparse adaptive sampling of the parametric space. This allows to recover the low-dimensional structure of the parametric solution subspace while also learning the functional dependency from the parameters in explicit form. A sparse low-rank approximate tensor representation of the parametric solution can be built through an incremental strategy that only needs to have access to the output of a deterministic solver. Non-intrusiveness makes this approach straightforwardly applicable to challenging problems characterized by nonlinearity or non affine weak forms. As we show in the various examples presented in the paper, the method can be interfaced with no particular effort to existing third party simulation software making the proposed approach particularly appealing and adapted to practical engineering problems of industrial interest.
Article
Full-text available
It is of great concern to produce numerically efficient methods for moisture diffusion through porous media, capable of accurately calculate moisture distribution with a reduced computational effort. In this way, model reduction methods are promising approaches to bring a solution to this issue since they do not degrade the physical model and provide a significant reduction of computational cost. Therefore, this article explores in details the capabilities of two model-reduction techniques - the Spectral Reduced-Order Model (Spectral-ROM) and the Proper Generalised Decomposition (PGD) - to numerically solve moisture diffusive transfer through porous materials. Both approaches are applied to three different problems to provide clear examples of the construction and use of these reduced-order models. The methodology of both approaches is explained extensively so that the article can be used as a numerical benchmark by anyone interested in building a reduced-order model for diffusion problems in porous materials. Linear and non-linear unsteady behaviors of unidimensional moisture diffusion are investigated. The last case focuses on solving a parametric problem in which the solution depends on space, time and the diffusivity properties. Results have highlighted that both methods provide accurate solutions and enable to reduce significantly the order of the model around ten times lower than the large original model. It also allows an efficient computation of the physical phenomena with an error lower than 10^{-2} when compared to a reference solution.
Article
The urban context is often simplified or neglected in Building Energy Models (BEMs) due to the difficulties of taking accurately into account all the heat fluxes emanating from the environment. Oversimplifying the urban context can impact the accuracy of the BEM predictions. Nevertheless, several approaches can be used to allow for the impact of the urban environment on the dynamic behavior of a building, its heating and cooling demands, and thermal comfort. This state of the art review provides a critical overview of the different methods currently used to take into account the urban microclimate in building design simulations. First, both the microclimate and building models are presented, focusing on their assumptions and capabilities. Second, a few examples of coupling, performed between both modeling scales are analyzed. Last, the discussion highlights the differences obtained between simulations that take the urban context into consideration and those that simplify or neglect urban heat fluxes. The remaining scientific obstacles to a more effective consideration of the urban context impacting the BEMs are indicated.
Conference Paper
A parametric soil model has been developed to improve the computational time of microclimate simulation tools. It combines the use of two methods: the Proper Generalized Decomposition and the Proper Orthogonal Decomposition. Offline, a learning process is required to build the model, before its use on-line. A methodology to select a short and representative learning process needs to be developed. The k-means clustering method is used to build a training climate made of 24 days representative of a full climate. The offline computation costs are reduced by 94.4% for an error of 0.8%.
Article
Numerical simulation is a powerful tool for assessing the causes of an Urban Heat Island (UHI) effect or quantifying the impact of mitigation solutions on local climatic conditions. However, the numerical cost associated with such a tool is quite significant at the scale of an entire district. Today, the main challenge consists of achieving both a proper representation of the physical phenomena and a critical reduction in the numerical costs of running simulations. This paper presents a combined parametric urban soil model that accurately reproduces thermal heat flux exchanges between the soil and the urban environment with a reduced computational time. For this purpose, the use of a combination of two reduced-order methods is proposed herein: the Proper Orthogonal Decomposition method, and the Proper Generalized Decomposition method. The developed model is applied to two case studies in order to establish a practical evaluation: an open area independent of the influences of the surrounding surface, and a theoretical urban scene with two canyon streets. The error due to the model reduction remains below 0.2 °C on the mean surface temperature for a reduced computational cost of 80%. Compared to in situ measurements the error remains bellow 1.24 °C at the surface.
Article
This work presents an efficient numerical method based on spectral expansions for simulation of heat and moisture diffusive transfers through multilayered porous materials. Traditionally, by using the finite-difference approach, the problem is discretized in time and space domains (Method of lines) to obtain a large system of coupled Ordinary Differential Equations (ODEs), which is computationally expensive. To avoid such a cost, this paper proposes a reduced-order method that is faster and accurate, using a much smaller system of ODEs. To demonstrate the benefits of this approach, tree case studies are presented. The first one considers nonlinear heat and moisture transfer through one material layer. The second case - highly nonlinear - imposes a high moisture content gradient - simulating a rain like condition - over a two-layered domain, while the last one compares the numerical prediction against experimental data for validation purposes. Results show how the nonlinearities and the interface between materials are easily and naturally treated with the spectral reduced-order method. Concerning the reliability part, predictions show a good agreement with experimental results, which confirm robustness, calculation efficiency and high accuracy of the proposed approach for predicting the coupled heat and moisture transfer through porous materials.
Article
One possibility to improve the accuracy of building performance simulation (BPS) tools is via co-simulation techniques, where more accurate mathematical models representing particular and complex physical phenomena are employed through data exchanging between the BPS and a specialized software where those models are available. This article performs a deeper investigation of a recently proposed co-simulation technique that presents as novelty the employment of artificial intelligence as a strategy to reduce the computational burden generally required by co-simulations. Basically, the strategy, known as intelligentco-simulation, constructs new mathematical models through a learning procedure (training period) that is performed using the input–output data generated by a standard co-simulation, where the models of specialized software are employed. Once the learning phase is complete, the specialized software is disconnected from the BPS and the simulation goes on by using the synthesized models, requiring a much lower computational cost and with a low impact on the accuracy of the results. The synthesis of accurate-and-fast models is performed through machine learning techniques and the purpose of this paper is precisely a deep investigation of two techniques – recurrent neural networks and proper orthogonal decomposition reduction method, whose main goal is to reduce the training time period and to improve the accuracy. The case study focuses on a co-simulation between Domus and CFX programs, performing a two-dimensional diffusive heat transfer problem through a building envelope. The results show that for a standard co-simulation of 14 h, the intelligent co-simulation provided a reduction of 90% in the computer run time with accuracy error at the order of .
Article
This work presents an alternative view on the numerical simulation of diffusion processes applied to the heat and moisture transfer through multilayered porous building materials. Traditionally, by using the finite-difference approach, the discretization follows the Method Of Lines (MOL), when the problem is first discretized in space to obtain a large system of coupled Ordinary Differential Equations (ODEs). This paper proposes to change this viewpoint. First, we discretize in time to obtain a small system of coupled ODEs, which means instead of having a Cauchy (Initial Value) Problem (IVP), we have a Boundary Value Problem (BVP). Fortunately, BVPs can be solved efficiently today using adaptive collocation finite-difference methods of high order. To demonstrate the benefits of this new approach, three case studies are presented. The first one considers nonlinear heat and moisture transfer through one material layer. The second case includes the rain effect, while the last one considers two material layers. Results show how the nonlinearities and the interface between materials are easily treated, by reasonably using a fourth-order adaptative method. In our numerical simulations, we use adaptive methods of the fourth order which in most practical situations is more than enough.
Article
This paper proposes the use of a Spectral method to simulate diffusive moisture transfer through porous materials as a Reduced-Order Model (ROM). The Spectral approach is an a priori method assuming a separated representation of the solution. The method is compared with both classical Euler implicit and Crank-Nicolson schemes, considered as large original models. Their performance - in terms of accuracy, complexity reduction and CPU time reduction - are discussed for linear and nonlinear cases of moisture diffusive transfer through single and multi-layered one-dimensional domains, considering highly moisture-dependent properties. Results show that the Spectral reduced-order model approach enables to simulate accurately the field of interest. Furthermore, numerical gains become particularly interesting for nonlinear cases since the proposed method can drastically reduce the computer run time, by a factor of 100, when compared to the traditional Crank-Nicolson scheme for one-dimensional applications.