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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 Ω τwith t ∈0, τ and space interval Ω xwith x ∈0,L:

c∂u(x,t)

∂t=∂

∂xk∂u(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:

−k∂u(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:

k∂u(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

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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 Ω Γ=0,Γand space interval Ω x=0,1, and the boundary condition as:

∂u(x, t)

∂x =Biout u−uout −q , x = 0 ,(6a)

∂u(x, t)

∂x =−Biin u−uin , x = 1 .(6b)

The initial condition becomes:

u= 0, t = 0 .(7)

Where the dimensionless quantities are deﬁned as:

u: = u−u0

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 y≡un+1 , Eq. (8) can be reformulated as:

y−a·∂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

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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 ·y−bout , 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×Ωbout ×Ωbin ×Ωζj−→ R,

x , b out , b in , ζ j7−→ ux , b out , b in , ζ j.

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The sets Ω bout , Ω bin and Ω ζjare the domain of variations of the coordinates bout ,bin and ζj, respectively. They are

deﬁned such as:

Ωbout =b−

out , b +

out ,Ωbin =b−

in , b +

in ,Ωζj=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.

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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 : Ωx=−1,1,

Ωbin =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):

ZΩx×Ωbin ×Ωbout ×Ωζj

y∗.

y−a∂2y

∂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.

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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=ζj−min (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.

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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.

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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 ℓ∞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

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

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.m−1.K−1and a speciﬁc heat capacity c = 2.2 10 6J.m−3.K−1.

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 [h−1], δ o, 2=−11.7 [ K], ω o, 2=1

24 [h−1],

qm= 500 [ W.m−2], ω q, =1

48 [h−1],ui,m= 20 [ ◦C], δ i=−2.0 [ K], ω i,1=1

48 [h−1].

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.m−2.K−1and h out = 23.2W.m−2.K−1.

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

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= 10−3and 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(N−7).

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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

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= 10−2,10−4,10−6. 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= 10−2, the threshold is reached for

N= 5 and 7 and 12 modes for respectively: ∆ζ= 10−4and ∆ζ= 10−6.

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∈[10−5,10−2]. 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= 10−6and ǫ= 10−8. The inﬂuence of each

parameter ∆ζ, N, and Mis studied hereafter, based on the results of those basis.

14 /32

2 3 4 5 6 7

10-4

10-3

10-2

(a) Evolution of the error for ∆ζ= 10−2

2 4 6 8 10

10-6

10-5

10-4

10-3

10-2

(b) Evolution of the error for ∆ζ= 10−4

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= 10−6

Figure 9. Inﬂuence of the truncation of the approximation basis on the accuracy of the approximation for various

discretizations

15 /32

(a) ε∞error 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= 10−5for 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= 10−4or ∆ζ= 10−5). 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(10−3) 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 10−3, the discretization of the boundary condition on x = 0 ﬁxed at 10−4, the spatial grid

ﬁxed at 10−2.

16 /32

(a) For ∆ζ= 10−5, evolution of the εerror (b) For ∆ζ= 10−5, 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 = 10−4and 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= 10−4for

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

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∈[10−5,10−2]

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= 10−6and ǫ= 10−8) as we increase the number of parameters involved.

(a) For δ= 10−4, 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

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

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=±0.1◦C. 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

5◦Cin 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 ∆ = 10−2) 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.m−1.K−1and c= 30.103J.m−3.K−1.

20 /32

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

5.1.4 Learning process

Three training data-sets for the POD basis are compared:

1. the full evaluation data-set (noted t∈Ωτ= [0 , τ] with τ= 9h),

2. half of the evaluation data-set, made of the cycles 0 to 3 (noted t∈Ωτ

2= [0 ,4h50]),

3. the cycle 1 (noted t∈Ωt1= [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].

2345678

10-6

10-5

10-4

10-3

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.

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Learning period ρCP U

t∈Ωτ1

t∈Ωτ

20.89

t∈Ωt10.65

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 (t∈Ωτ) and unfavourable (t∈Ωt1) 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∈[10−5,10−4]. 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= 10−6and ǫ= 10−8. The accuracy of the PGD parametric model is compared for various number of modes

N ∈ [2,5] and for a ﬁxed discretization ∆ζ= 10−5. We use the same parameters than in section 4.3.2 to compare the

results.

(a) For ∆ζ= 10−5, evolution of the εerror (b) For ∆ζ= 10−5, 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(10−3)) 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

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= 10−5are 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

(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

(a) For ∆ζ= 10−5, evolution of the εerror at 4cm from the inner

boundary condition

(b) For ∆ζ= 10−5, 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

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:

k∂eu

∂x =−hin eu−uin +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

:= u−eu . (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 e−qin , 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

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 =fwǫwσ4T4−u4

w+fgǫgσT4−u4

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.m−2.K−4, 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.m−2. 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

1.0◦Clocated, 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 0.2◦C. On the inside surface, the discrepancy between the solution is higher, around 0.5◦C.

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=2x2−1, T 3=4x3−3x.(27)

They are constructed according to the following relation of recurrence [5,19]:

Tj+ 1 = 2 x T j−Tj−1for j > 1 with T0= 1,and, T1=x(28)

The Chebyshev approximation basis is made of the Chebyshev polynomials.

Ψj≡Tj(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

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

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

2x2−1

2.(31)

The next polynomials are constructed according to the following relation of recurrence [5]:

(j+ 1 ) Pj+ 1 = ( 2 j+ 1 ) x P j−j P j−1for j≥1,and P0x= 1 , P 1x=x(32)

The Legendre approximation basis is made of the Legendre polynomials calculated at the Legendre points.

Ψj≡Pj(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).

Ψj≡Φj(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.

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