Content uploaded by Mickaël Rivier
Author content
All content in this area was uploaded by Mickaël Rivier on Sep 12, 2019
Content may be subject to copyright.
ISSN 0249-6399 ISRN INRIA/RR--9175--FR+ENG
RESEARCH
REPORT
N° 9175
September 12, 2019
Project-Teams CARDAMOM
Ablative thermal
protection system under
uncertainties including
pyrolysis gas composition
Mickaël Rivier, Jean Lachaud , Pietro Marco Congedo
RESEARCH CENTRE
BORDEAUX – SUD-OUEST
351, Cours de la Libération
Bâtiment A 29
33405 Talence Cedex
Ablative thermal protection system under
uncertainties including pyrolysis gas
composition
Mickaël Rivier∗†, Jean Lachaud ‡, Pietro Marco Congedo §
Project-Teams CARDAMOM
Research Report n°9175 — September 12, 2019 — 22 pages
Abstract: Spacecrafts such as Stardust (NASA, 2006) are protected by an ablative Thermal
Protection System (TPS) for their hypersonic atmospheric entry. A new generation of TPS mate-
rial, called Phenolic Impregnated Carbon Ablator (PICA), has been introduced with the Stardust
mission. This new generation of low density carbon-phenolic composites is now widely used in the
aerospace industry. Complex heat and mass transfer phenomena coupled to phenolic pyrolysis and
pyrolysis gas chemistry occur in the material during atmospheric entry. Computer programs, as
the Porous material Analysis Toolbox based on OpenFoam (PATO) released open source by NASA,
allow to study the material response. In this study, a non-intrusive Anchored Analysis of Variance
(Anchored-ANOVA) method has been interfaced with PATO to perform low-cost sensitivity anal-
ysis on this problem featuring a large number of uncertain parameters. Then, a Polynomial-Chaos
method has been employed in order to compute the statistics of some quantities of interest for the
atmospheric entry of the Stardust capsule, by taking into account uncertainties on effective mate-
rial properties and pyrolysis gas composition. This first study including pyrolysis gas composition
uncertainties shows their key contribution to the variability of the quantities of interest.
Key-words: Sensitivity analysis, Uncertainty quantification, Porous media, Heat-shield ablation
∗Inria Bordeaux Sud-Ouest - Team CARDAMOM
†ArianeGroup, Le Haillan
‡C la Vie, University of New Caledonia, New Caledonia
§DeFI - CMAP - Ecole Polytechnique, Inria Saclay - Ile de France, Polytechnique - X, CNRS
Système de protection thermique ablatif sous incertitudes
avec prise en compte des gazs de pyrolyse
Résumé : Les véhicules spaciaux tels que Stardust (NASA, 2006) sont protégés par un sys-
tème de protection thermique ablatif (TPS) lors de leur entrée atmosphérique hypersonique.
Une nouvelle génération de matériau de protection, appelé Phenolic Impregnated Carbon Abla-
tor (PICA), a été introduite pour la mission Stardust. Cette nouvelle génération de composites
carbone-phénolique à faible densité est maintenant couramment utilisée dans l’industrie aérospa-
tiale. Les phénomènes complexes de transfert de masse et de chaleur couplés à la pyrolyse de la
matrice phénolique et la chimie des gazs associés surviennent dans le matériau durant l’entrée
atmoshpérique. Différents codes de calcul, comme Porous material Analysis Toolbox based on
OpenFoam (PATO) mis en accès libre par la NASA, permettent d’étudier la réponse du matériau.
Dans notre étude, une méthode non-intrusive Anchored Analysis of Variance (Anchored-ANOVA)
a été interfacée avec PATO pour réaliser une analyse de sensibilité à faible coût sur ce problème
comportant un grand nombre de parmètres incertains. Ensuite, une méthode de chaos polyno-
mial a été mise en oeuvre afin de calculer les statistiques de différentes quantités d’intérêt lors de
l’entrée atmosphérique de la capsule Stardust, en considérant les incertitudes sur les propriétés
matériau et la composition des gazs de pyrolyse. Cette première étude prenant en compte les
incertitudes sur la composition des gazs de pyrolyse montre leur forte contribution à la variabilité
finale des quantités d’intérêt.
Mots-clés : Analyse de sensibilité, Quantification des incertitudes, Milieu poreux, Abalation
de bouclier thermique
Uncertainty propagation for heat and mass transfer in porous media 3
1 Introduction
Space exploration missions often include entering a planetary atmosphere at hypersonic speed. A
high enthalpy hypersonic shock forms around the spacecraft and kinetic energy is progressively
dissipated into heat [1]. Heat is transferred to the surface of the spacecraft by radiation and
convection. A suitable heat shield is needed to protect the payload. The level of heat flux
increases with entry speed and atmospheric density. For fast hypersonic entries, typically faster
than 8km/s from earth orbit, ablative materials are used as Thermal Protection Systems (TPS).
These materials mitigate the incoming heat through phase changes, chemical reactions, and
material removal [2]. A low-density porous carbon/phenolic composite called PICA was used
for the Stardust comet-dust sample-return capsule, which reentered the Earth’s atmosphere at
12.7km/s [3]. PICA is made of a carbon fiber preform partially impregnated with phenolic resin.
Virgin&
material&
Pyrolysis&
zone&
Abla3on&&
zone&
&
!"#$$"%"
!"&'$$"%"
!"&#$$"%"
Coking&
zone&
!"($$$"%"
OH
Phenolic polymer
H2 +
C6H6 + H2O
C6 + 3 H2
3 C2H2
C (s)
2 O
O2
2 C (s) + O2
! 2 CO
)*+,-./01",+2*34-.,."
5.-,67-8+9"-77:./03;<4="
>2344-4?"@7+2/0<4"A-20<.2<61"5>@A="<B"
3"230C<4"60+B<0,"C+B<0+"6*+4<7-2"
-,60+?43;<4"
>@A"<B"DE)F"5G-0?-4="H>/32I6<<7+J"'$$KL"
Boundary&
layer&
Non:viscous&
flow&
Porous microstructure
M7<NO.:0B32+"-4/+032;<4."
P"FC73;<4"5<Q-93;<4J".:C7-,3;<4J".63773;<4=J""
P"R+3/"349",3..".:0B32+"C37342+"5&="
"
>:C.:0B32+"6*+4<,+43"54</"37N31.="
P "S<:49301"731+0"?3.+."+4/+0-4?"-4"/*+",3/+0-37"5(="
P "E4P9+6/*"<Q-93;<4"5(="
R+3/"349",3.."/034.6<0/"
"P"><7-9"9+4.-/1"2<4.+0G3;<4"5&="
"P">/+391P./3/+"?3."2<4.+0G3;<4"5&="
P""T3."9+4.-/1"2<4.+0G3;<4"5'="
"P"T3.",<,+4/:,"2<4.+0G3;<4"5'="
"P"T3."+7+,+4/O.6+2-+."2<4.+0G3;<4"5(="
"P"@4+0?1"2<4.+0G3;<4"5&="
D0<9:2;<4"6*+4<,+43"
P"D10<71.-."60<9:2;<4"03/+"5&="
P"D10<71.-.".6+2-+.O+7+,+4/."60<9:2+9"5(="
P")*+,-./01"<B"/*+"?3.+."5(="
"
U<4?"9-./342+"+V+2/."
P"W39-3;<4"
2<I-4?"
039X"
R+3/"
!"Y$$$"%"
&"Z"-4"377",3/+0-37P0+.6<4.+",<9+7."5/16+"&="
'"Z"-4".<,+",3/+0-37P0+.6<4.+",<9+7."5/16+"'="
("Z"-4"9+/3-7+9",3/+0-37P0+.6<4.+",<9+7."5/16+"(="
A3/+0-37"0+.6<4.+"5DF[\=" R16+0.<4-2"]<N"
.*<2I"
Figure 1: Phenomenology of porous carbon/phenolic ablative materials
During atmospheric entry, carbon/phenolic materials undergo thermal degradation and ulti-
mately recession captured by the following physico-chemical phenomena (Figure 1). The pheno-
lic polymer thermally decomposes and progressively carbonizes into a low density carbon form,
losing mass while releasing pyrolysis gases. The pyrolysis gases percolate and diffuse to the
surface through the network of pores. Reactions within the pyrolysis-gas mixture (homogeneous
reactions) and between pyrolysis gases and the char take place with possible coking effects (het-
erogeneous reactions). Mixing and reaction of the pyrolysis gases with boundary layer gases into
the pores of the material occur when boundary layer gases penetrate in the material by forced
RR n°9175
4Rivier, Lachaud, Congedo
convection or due to fast diffusion at low pressures [4]. At the surface, the material is removed by
ablation and the outer surface recedes. Depending on entry conditions, ablation may be caused
by heterogeneous chemical reactions (oxidation, nitridation), phase change (sublimation), and
possibly mechanical erosion (often called spallation).
A detailed heat and mass transfer model is required to estimate the performance of the
porous material and design the thermal protection system. Two important design criteria are
the expected level of recession and the maximum back wall temperature. The key parameter
uncertainties are propagated to obtain the design uncertainties to be used in the margin policy
[5]. A Monte Carlo approach has been developed and used to propagate uncertainties on material
properties and areoheating conditions for the design of the NASA Mars Science Laboratory [6]
and Orion [7] spacecrafts. In these study however, no uncertainty is attributed to the pyrolysis
gas composition. Recent publications have shown that the pyrolysis gas composition strongly
varies depending on temperature and heating rate [8, 9, 10]. The pyrolysis gases are composed
of carbon, oxygen and hydrogen elements. The pyrolysis gas composition influences the pyroly-
sis gas enthalpy - which impacts heat transfer in the porous material - and the boundary layer
chemistry - which controls the ablation rates and the surface temperature. For this first analysis
we will allow an uncertainty of 10% on these elements. The composition in term of species is
then computed in each cell of the mesh and at each time step using an equilibrium chemistry
solver [4]. This makes the computation very costly and requires the use of the low-cost un-
certainty quantification methods. In the literature, low-cost uncertainty propagation has been
already performed alongside Global Sensitivity Analysis for problems of natural convection in
[11]. Uncertainty analyses have also been performed on surface ablation rates and their effect on
aeroheating predictions for Mars entry in [12], and on ablation problems in plasma wind tunnel
[13, 14].
In section 2, we present the problem studied and the physical hypotheses. In section 3, we
present the inverse analysis method implemented in the study. The results of the uncertainty
quantification analysis are presented in section 4. Finally, section 5 draws some conclusions and
perspectives.
2 Definition of the uncertainty analysis problem and hy-
potheses
For this first analysis, we chose to study the entry of Stardust, that was the first mission using
a low-density carbon-phenolic ablator in 2006. The thermal response of the TPS has been be
studied at the stagnation point during the whole reentry, from entry interface to cool down. As in
the state-of-the-art design approach we assumed that the problem is locally mono-dimensional.
The actual thickness of the ablative material was two inches [3], therefore we used this value.
Adiabatic conditions are used at the bondline. A convective boundary condition is used at the
surface of the ablative material. Surface total pressure and heat flux were taken from reference
[15].
Figure 2 illustrates the temperature evolution during the atmospheric entry at the stagnation
point and in-depth under the stagnation point with nominal TACOT properties.
The analysis is performed using the properties of the Theoretical Ablative Composite for Open
Testing (TACOT). Its composition and properties are comparable to PICA. Nominal TACOT
properties are available in the open literature [16]. Volume-wise, TACOT is made of 10% of
carbon fibers, 10% of phenolic resin, and is 80% porous.
Inria
Uncertainty propagation for heat and mass transfer in porous media 5
Figure 2: Surface and in-depth temperatures obtained with nominal material and pyrolysis gas
composition parameters.
2.1 Model
A generic heat and mass transfer model for porous media has been recently developed and doc-
umented [17]. It is suitable to model ablative heat shields. For the sake of conciseness, we only
present a short summary in this section. The model was developed for porous materials con-
taining several solid phases and a single gas phase. The detailed chemical interactions occurring
between the solid phases and the gas phase are modeled at the pore scale assuming local thermal
equilibrium. Homogenized models were obtained for solid pyrolysis, pyrolysis species injection
in the gas phase, heterogeneous reactions between the solid phases and the gas phase, and ho-
mogeneous reactions in the gas phase. The chemistry models were integrated in a macroscopic
model making use of volume-averaged governing equations for the conservation of solid mass,
gas mass, species (finite-rate chemistry) or elements (equilibrium chemistry), momentum, and
energy. The model is implemented in the Porous material Analysis Toolbox (PATO), distributed
Open Source by NASA. First-order implicit finite-volume schemes in time and space [18, 19],
which have been shown to provide excellent convergence and accuracy [20, 4], were used for the
simulations presented in Section 4. In this study, we used an equilibrium chemistry model that
is equivalent to the reference NASA TPS design model [21]. The current approach is to assume
that the elemental pyrolysis gas composition is fixed. To save on computational time, precom-
puted tables are used to obtain the gas composition (species) and properties (enthalpy, viscosity,
RR n°9175
6Rivier, Lachaud, Congedo
molar mass). In the current study, we wish to vary the elemental pyrolysis gas composition. The
pyrolysis gas composition in term of species is therefore computed from pyrolysis gas elemental
composition in each cell of the mesh and at each time step using an equilibrium chemistry solver,
as described in [4]. The surface ablation rate is computed using the thin film coefficient approach,
also known as B’ approach, accounting for the change in pyrolysis gas composition injected in
the boundary layer [4].
2.2 Uncertain parameters and associated uncertainties
In previous studies, uncertain material property parameters have been identified for PICA [6,
5, 22]. We decided to include the same uncertain parameters in our study. We also added
a set of new parameters to assess the effect of the pyrolysis gas composition on the material
response as described in the introduction. The nominal elemental composition of the pyrolysis
gases for TACOT are, in mole fractions, C (0.206), H (0.679), O (0.115). In total, we have used
twenty-seven uncertain parameters in the TACOT material model. We have attributed 5 to 10
% uncertainty to each of them as follows (the number in brackets is the label used to identify
each uncertainty in the following of this paper):
•Density (1) and volume fraction (2) of the fibrous preform (5% uncertainty),
•Density (3) and volume fraction (4) of the phenolic matrix (5% uncertainty),
•Virgin’s (5) and char’s (6) permeability (5% uncertainty),
•Pyrolysis model (10% uncertainty):
–Elementary composition of the pyrolysis gases in Carbon (7) , Hydrogen (8) and
Oxygen (9),
–Pyrolysis reaction 1: pre-exponential factor (10), activation energy (11), pyrolysis
enthalpy (12),
–Pyrolysis reaction 2: pre-exponential factor (13), activation energy (14), pyrolysis
enthalpy (15),
•Thermal properties of virgin material (5% uncertainty): heat capacity (16), orthogonal
conductivity (17), radial conductivities (18, 19), emissivity (20), reflectivity (21),
•Thermal properties of charred material (5% uncertainty): heat capacity (22), orthogonal
conductivity (23), radial conductivities (24, 25), emissivity (26), reflectivity (27).
We chose to simplify the constraint of elementary fractions summing to one through maintain-
ing the relative ratios and normalizing the elementary composition. Practically, given a random
draw of the mass fractions yiwithin the ±10% interval, normalized mass fractions eyi=yi
P
k
yk
will actually be given to PATO.
Moreover, with 1and 2the uncertain volume fraction of the fibrous preform and the virgin
matrix respectively, the virgin and charred porosities εvand εcare computed as follows:
εv= 1 −1−2
εc= 1 −1−2
2
Finally, one may note the presence of 3D conductivities in the list of uncertain parameters,
which clashes with the mono-dimensional assumption made earlier. Such parameters are left in
Inria
Uncertainty propagation for heat and mass transfer in porous media 7
the study to artificially increase the input dimension and verify the capability of the proposed
approach to detect their null impact and discard them.
3 Sensitivity and uncertainty analysis theory and tools
Let us consider a stochastic differential equation of the form:
L(x,ξ, φ) = f(x,ξ)(1)
where Lis a non-linear spatial differential operator (for instance, the steady Navier-Stokes oper-
ator) depending on a set of uncertainties, designated with the random vector ξ(of dimension the
number of uncertain parameters in the problem) and where f(x,ξ)is a source term depending
on xand ξ. In the following, we drop the dependence on xin order to simplify the notation.
The solution of the stochastic equation (1) is φ(ξ), which is a function of the space variable
x∈Rdand of ξ∈Ξ = Ξ1× ·· · × ΞN(Ξ⊂RN) and ξ∈Ξ7−→ φ(ξ)∈L2(Ξ, p(ξ)), where
p(ξ) = QN
i=1 p(ξi)is the probability density function of ξ.
One of the objective of Uncertainty Quantification is to compute the statistics of the quantity
of interest, i.e. φ(ξ).
We can define the central statistical moment of φof order nas
µn(f) = ZΞ
(φ(ξ)−E(φ))np(ξ)dξ, (2)
where E(φ)indicates the expected value of φ
E(φ) = ZΞ
φ(ξ)p(ξ)dξ. (3)
In the following, we indicate with σ2, the variance (second-order moment). We illustrate the
main concepts of the ANOVA-decomposition in Section 3.1. Then, to clearly present the context
of uncertainty analysis theory and provide a comprehensive understanding of the approach fol-
lowed in this work, we will use as illustration a mono-dimensional heat transfer problem presented
in Section 3.2. The UQ methods are then described in Section 3.3.
3.1 ANOVA-based decomposition
Let us suppose that the response of a given system of interest can be represented by a N−dimensional
function:
y=φ(ξ) = φ(ξ1,ξ2,·· · ,ξN)(4)
We consider Eq. (4) in its functional expansion form as follows
y=φ0+
N
X
16i6N
φi(ξi) +
N
X
16i<j6N
φij (ξi,ξj) + · ·· +φ1,2,··· ,N (ξ1,ξ2,··· ,ξN)
or in compact form using a multi index system:
y=φs0+
2N−1
X
j=1
φsj(ξsj)(5)
RR n°9175
8Rivier, Lachaud, Congedo
The multi indices sjare defined such as
s0= (0,0,0,·· · ,0)
s1= (1,0,0,·· · ,0)
s2= (0,1,0,·· · ,0)
.
.
.
sN= (0,0,0,·· · ,1)
sN+1 = (1,1,0,·· · ,0)
sN+2 = (1,0,1,·· · ,0)
.
.
.
sN= (1,1,1,·· · ,1)
(6)
where N= 2N−1. The representation of Eq. (5) is called ANOVA (Analysis Of Variance)
decomposition [23] of φ(ξ), if for any j∈ {1,·· · ,N},
ZR
φsj(ξsj)p(ξi) dξi= 0 for ξi∈ {ξsj}(7)
It follows from Eq. (7) the orthogonality of ANOVA component terms, namely
E(φsjφsk)=0 for j6=k(8)
ANOVA allows identifying the contribution of a given stochastic parameter to the total variance
of an output quantity. Meanwhile, we obviously have
E(φsj)=0 for j= 1,·· · ,N
Note that the terms in the ANOVA decomposition can be expressed as integrals of φ(ξ). Indeed,
we have E(Y) = φ0
E(Y|ξi) = φ0+φi(ξi)
E(Y|ξi,ξj) = φ0+φi(ξi) + φj(ξj) + φij (ξi,ξj)
(9)
and so on, where E(Y|·)denotes the conditional expectation.
3.2 Analytical solution for transient heat transfer
3.2.1 Deterministic problem
Effective heat transfer is the main mode of energy transport in most porous materials. Let us
consider a homogeneous semi-infinite unidimensional medium. Under the assumption of constant
material properties, the transient heat transfer equation is given by
∂tT=α ∂2
xT(10)
where α=k/(ρ×cp)is the diffusivity. We will consider a medium initially at the room temper-
ature T(x, t = 0) = T0= 300 K. Its surface temperature is held at T(x= 0, t) = Tw= 1646 K
during the experiment. Laplace transform is used for its resolution as presented in Appendix A.
The temperature profile as a function of time and space is given by
T(x, t) = T0+ (Tw−T0)erfc x
2√αt(11)
Inria
Uncertainty propagation for heat and mass transfer in porous media 9
where erfc is the complementary error function. The temperature profiles computed for times
of 1, 10 and 60 seconds are plotted in Fig. 3 for a representative medium of diffusivity of 10−7
m2/s. We will only consider in the illustrations that follow the first centimeter of the medium.
We see here that the hypothesis that the medium is semi-infinite does not play a role on the
result, as the heat wave hasn’t reached the one centimeter mark after one minute of heating.
Figure 3: Analytical solution for unidimensional transient heat transfer with fixed surface tem-
perature, for a semi-infinite medium of thermal diffusivity 10−7m2/s. Initial temperature of the
body: 300 K; surface temperature: 1646 K. Left: temperature profiles for 1, 10, and 60 seconds.
Right: error bars in terms of standard deviation when considering two uncertainties.
3.2.2 Formulation under uncertainty
Let us now formulate the problem presented in Eq. 10 under an uncertainty quantification
perspective. In particular, let assume that two parameters are affected by some variability
and/or are not well-known: a 5% of variation in terms of min/max is then imposed on Twand
αconsidering a uniform distribution (with respect to the deterministic values previously used,
denoted in the following as Twm and αm, respectively). The problem is now formulated as follows:
∂tT=α ∂2
xT=αm×(0.95 + (1.05 −0.95) ∗ξ2)∂2
xT(12)
where Tw=Twm ×(0.95 + (1.05 −0.95) ∗ξ1), and ξ1,ξ2vary in [0,1].
Several methods can be used in order to solve the problem defined in Eq. 12. In this work,
we use systematically the so-called non-intrusive methods: this means that a single deterministic
computation (used to solve for example the differential operator defined in Eq. 10) is replaced
with a whole set of such computations, each one of those being run for specific values of the
uncertain conditions. The final solution can be then written as follows:
T(x, t, ξ1,ξ2) = T0+ (Twm(0.95 + 0.1ξ1)−T0)erfc x
2pαm(0.95 + 0.1ξ2)t!.(13)
RR n°9175
10 Rivier, Lachaud, Congedo
Now, let us show how the computation of the variance and the computation of the contribution
of each source of uncertainty can be reduced only to the computation of some integrals on the
analytical solution shown in Eq. 13 for some fixed values of xand t:
The ANOVA functional expansion (more details are provided in the next subsection) is a
unique tool for assessing the contribution of each uncertainty (and of the interactions) to the
global variance. This is computed as follows (variables x and t are dropped since this does not
change the following developments)
T(ξ) = T0+Tξ1+Tξ2+Tξ1ξ2,(14)
where
T0=ZΞ2
T(ξ)p(ξ)dξ;
Tξ1=ZΞ
T(ξ)p(ξ2)dξ2−T0;
Tξ2=ZΞ
T(ξ)p(ξ1)dξ1−T0;
Tξ1ξ2=T(ξ)−Tξ1−Tξ2−T0.
(15)
The overall variance σ2can be computed by means of the ANOVA expansion as
σ2=σ2
ξ1+σ2
ξ2+σ2
ξ1ξ2,(16)
where
σ2
ξ1=ZΞ
T2
ξ1p(ξ1)dξ1;
σ2
ξ2=ZΞ
T2
ξ2p(ξ2)dξ2;
σ2
ξ1ξ2=ZΞ2
T2
ξ1ξ2p(ξ)dξ.
(17)
Note that σ2
ξ1,σ2
ξ2represent the unique contribution of ξ1and ξ2to the global variance σ2,
respectively. Moreover, σ2
ξ1ξ2represents the contribution given by the interaction between ξ1
and ξ2.
Note that only integrals of the expression defined in Eq. 13 are required, in order to compute
the contributions to the variance for fixed values of (x, t). In Figure 3 (on the right), the solution
is then represented in terms of mean and the associated error bars (square root of the variance,
i.e. standard deviation). Figure 4 illustrates the variance of the temperature t= 60 s, induced
by each uncertainty. Note that the contribution Twis predominant and explains most of the
global variability of the temperature. This simple example illustrates the interest in propagating
some physical input uncertainties through numerical models.
3.3 Non-intrusive formulations for expensive computer codes
Unfortunately, generally, it is not possible to compute an analytical solution of the problem
defined in Eq. 1. This could require the resolution of a complex system of equation, relying on a
numerical approximation of the solution on some discretized grid of the numerical domain. Note
then that computing the integrals of Eq. 17 can be very costly. Moreover, some additional issues
Inria
Uncertainty propagation for heat and mass transfer in porous media 11
Figure 4: Variance of the temperature (including the contribution of each uncertainty) at a time
of 60 seconds.
could arise in the presence of a large number of uncertainties or if the quantity of interest features
some discontinuities. The real challenge is then to formulate an efficient numerical algorithm
permitting to build an accurate representation of the quantity of interest as a function of input
uncertainties.
As previously mentioned, only non-intrusive strategy are targeted in this work. In particular,
here, we tackle a problem featuring a large number of uncertainties, that can be very challenging
to solve, due to the so-called Curse of Dimensionality. It refers to the loss of convergence and the
infeasible number of calculations needed when the number of parameters increases, for any chosen
method. We have partially cured this problem with a two-steps approach. First, we applied an
anchored-ANOVA approach on the complete problem. This analysis permits to compute the
hierarchy and detect the most important uncertainties. Note that this approach only needs
a very reduced number of deterministic simulations to perform uncertainty propagation and
sensitivity analysis. In a second step, we applied a Polynomial-Chaos approach for treating the
subspace including only the predominant parameters, in order to provide a good representation
of the quantity of interest in the reduced stochastic space.
As mentioned before, due to the non-intrusivity of the stochastic methods considered here,
the coupling with PATO, or any other heat and mass transfer computational model, is very
straightforward: it reduces to the creation of a small interface for building automatically PATO
input parameters files for each set of uncertain conditions.
Both methods are described briefly in the following. For more details, refer to [24] for the
Polynomial-based method and to [25] for the anchored-ANOVA approach.
RR n°9175
12 Rivier, Lachaud, Congedo
3.3.1 Anchored-ANOVA approach: Definitions and basic notions
In order to introduce the less expensive anchored ANOVA, the Dirac measure is used for the
integrals of Eq. (9):
p(ξi) dξi=δ(ξi−ci) dξifor i= 1,·· · , N (18)
Thus, p(ξ) dξ=δ(ξ−c) dξ. The point c= (c1,··· , cN)is called “anchor point”. Hence, the
ANOVA component terms in Eq. (9) can be expressed as follows:
φ(c) = φ0
φ(c|ξi) = φ0+φi(ξi)
φ(c|ξi,ξj) = φ0+φi(ξi) + φj(ξj) + φij (ξi,ξj)
.
.
.
(19)
The formulae in Eq. (19) are used to quantify the expectation and variance of the compo-
nent functions, by simply evaluating the model outputs at chosen sampling points. For more
details, see [25]. This permits a strong reduction of the computational cost, since this avoids the
computation of several integrals. Moreover, a variance-based adaptive criterion (see for more
details [26]) is used in order to compute the so-called effective dimension and to evaluate high-
order interactions with a reduced computational cost. The order at which the ANOVA model is
truncated, is called effective dimension, beyond which the difference between the ANOVA model
and the truncated expansion in a certain measure is very small. This implies that we will ignore
terms in the ANOVA model corresponding to interactions exceeding the fixed threshold.
In this work, a covariance decomposition of the output variance has been considered, as
proposed in [25], in order to accurately compute the statistics using the anchored-ANOVA ex-
pansion. The covariance decomposition makes the result less sensitive to the choice of the anchor
point if a full expansion of the anchored ANOVA is employed.
3.3.2 Polynomial-chaos based approach
Under specific conditions, a stochastic process can be expressed as a spectral expansion based
on suitable orthogonal polynomials, with weights associated to a particular probability density
function. The first study in this field is the Wiener (1938) process. The basic idea is to project
the variables of the problem onto a stochastic space spanned by a complete set of orthogonal
polynomials Ψthat are functions of random variables ξ. For example, the unknown variable φ
has the following spectral representation:
φ(ξ) =
∞
X
i=0
φiΨi(ξ)) .(20)
In practice, the series in Eq. (20) has to be truncated in terms of the polynomial degree p0,
where the total number of terms of the series Mis determined by:
M+ 1 = (N+p0)!
N!p0!,(21)
where Nis the dimensionality of the uncertainty vector ξ. Each polynomial Ψi(ξ)is a multivari-
ate polynomial form which involves tensorization of 1D polynomial forms. The polynomial basis
is chosen accordingly to the Wiener-Askey scheme [27] in order to select orthogonal polynomials
Inria
Uncertainty propagation for heat and mass transfer in porous media 13
with respect to the probability density function p(ξ)of the input. The orthogonality can be ad-
vantageously used to compute the coefficients of the expansion in a non-intrusive PC framework
φi=hφ(ξ),Ψi(ξ)i
hΨi(ξ),Ψi(ξ)i,∀i. (22)
Several approaches can be used to estimate PC coefficients. The approach used in this study
is based on quadrature formula. As a consequence, the solution of a deterministic problem for
each quadrature point is required.
For further details, see Congedo et al.[24]. In both cases, once the chaos polynomials and the
associated φicoefficients are computed, the expected value and the variance of the stochastic
solution φi(ξ)are obtained from:
EP C =φ0(23)
V arP C =
N
X
i=1
φ2
iΨ2
i(24)
Another interesting property of PC expansion is to make easier sensitivity analysis based on
the analysis of variance decomposition (ANOVA). It can be easily computed by using some
interesting properties of the previous development [28]. Let us recall here that the contribution
to the variance of a given random variables with index k,i.e. the first order Sobol’s index, can
be obtained by:
Sk=Pi∈αφ2
ihΨ2
i(ξ)i
V arP C
(25)
where αrepresent the set of indexes associated to a given uncertainty k. For more details, Ref.
[28] is strongly recommended.
4 Results
The Anchored-ANOVA method is applied to the problem presented in section 2. We present first
the convergence analysis with respect to the number of samples considered to get well-converged
statistics of all the quantities of interest. In Anchored-ANOVA, the first-order analysis is based
on a chosen number of points per direction. We have reported in Table 1 the outcome of this
analysis for the temperature computed at a depth of 1.5cm at a time equal to 80s, in terms
of decreasing contributions to the variance, for eight and sixteen points along each direction,
respectively (which makes a total number of runs of the solver of 216 and 432, respectively).
As it can be observed, errors are quite small, indicating that quantities are converged with only
eight points per direction. Same conclusions can be drawn for all the other quantities of interest
considered in this work, i.e. temperatures at different depths, the virgin front and the char front.
Results shown in the following rely then on this analysis.
Let us now analyse the results from a quantitative and qualitative point of view. Figure 5
presents the transformation of the ablative material with error bars, which represent the spread-
ing of the quantities of interest in terms of standard deviation. The plotted results are the
surface recession - due to ablation - and the propagation of the pyrolysis front. The char 2% and
virgin 98% are used in the ablation community to identify almost completely charred material
(2% left of virgin matrix) and almost pristine virgin material (98% left of virgin matrix). The
charring zone is considered to be between these two accepted limits [16]. Note that, since the
heating is quite soft at the beginning of the simulation, the material is fully pyrolysed before
being ablated. This explains why the char 2% and the wall curves are clearly separated in this
RR n°9175
14 Rivier, Lachaud, Congedo
Table 1: Contribution to the variance of each uncertainty and error analysis with respect to the
number of points per direction. Absolute and relative differences are also given.
Unc. Contrib. q16 (%) Contrib. q8 (%) ∆abs ∆r el
24 16.9 16.9 0.0 0.0
3 14.8 14.8 0.0 0.0
4 14.7 14.8 0.1 6.8·10−3
26 9.03 9.01 0.02 2.2·10−3
1 8.31 8.32 0.01 1.2·10−3
2 8.29 8.3 0.01 1.2·10−3
9 7.63 7.64 0.01 1.3·10−3
11 5.13 5.13 0.0 0.0
22 4.05 4.05 0.0 0.0
8 2.88 2.88 0.0 0.0
14 2.88 2.88 0.0 0.0
12 2.48 2.48 0.0 0.0
15 1.48 1.48 0.0 0.0
7 1.15 1.15 0.0 0.0
16 0.166 0.166 0.0 0.0
18 5.05·10−25.03·10−20.2·10−34.0·10−3
10 3.78·10−23.78·10−20.0 0.0
13 5.81·10−35.81·10−30.0 0.0
20 6.68·10−46.7·10−40.2·10−53.0·10−3
6 7.22·10−57.21·10−50.1·10−61.4·10−3
5 1.01·10−51.04·10−50.3·10−63.0·10−2
17 0.0 0.0 0.0 0.0
19 0.0 0.0 0.0 0.0
21 0.0 0.0 0.0 0.0
23 0.0 0.0 0.0 0.0
25 0.0 0.0 0.0 0.0
27 0.0 0.0 0.0 0.0
case. The observed variability is very small for the three curves. In fact, for the virgin 98%,
which is the worst case, the Coefficient of Variation (standard deviation to mean ratio) is of the
order of 3%.
A physical analysis of the results is now presented in terms of the contribution of each
uncertainty to several quantities, namely the recession, the virgin front, the char front and
the surface temperature, as a function of time. The results are presented in Figure 6. The
uncertainties propagated on the material properties are in agreements with previous studies
[6, 5, 22]. Concerning the recession and the location of the Char 2%, the maximal standard
deviation is of the order of 0.03-0.04 cm. As it can be observed in Figure 6a and b, the location
of the recession and char 2% mostly depend on both the parameters influencing the heat transfer
in the charred material and the pyrolysis parameters. The virgin 98% is predominantly influenced
by the uncertainty on the Activation energy 1, which controls the initiation of pyrolysis. Finally,
the surface temperature is mostly driven by the char’s emissivity except at the start of the
pyrolysis reaction, where the activation energies and gas composition are predominant.
We would like to point out a new result. The uncertainty on the elemental pyrolysis gas com-
position (Carbon, Hydrogen and Oxygen) clearly induces variability on the studied quantities.
The effect of the these uncertainties is observed on the in-depth temperature evolution as well,
as shown in Figure 7, where the contributions of each uncertainty to the variance of the tem-
perature is computed over the time at different depths in the material. The standard deviation
of the temperature takes the highest value at a depth of 0.7 cm (100 K), while it remains quite
small for the other depths considered here.
With the advancement of the ablation front, contributions of the virgin’s parameters are
quickly overtaken by char’s ones. Parameters of the pyrolysis reaction and material composition
also show decreasing contributions as the pyrolysis reaction comes to an end.
Inria
Uncertainty propagation for heat and mass transfer in porous media 15
Figure 5: Recession, char at 2% and virgin at 98% with the ±standard deviation envelope.
In order to make more evident the different contributions, Figure 8 illustrates the differ-
ent contributions gathered in terms of different groups of uncertainty, i.e. fibers and matrices
properties, TACOT’s composition, pyrolysis parameters, etc.
By using the low-cost sensitivity analysis technique, the hierarchy of the most important
uncertain parameters contributing to the variance of the temperature can be computed as a
function of the depths at a fixed time. For example, in Figure 9 we show the hierarchy at a time
of 80 s. As it can be observed, the trend is highly non linear, and this could be particularly
useful to build reliable design margin policies and to guide material model development efforts.
4.1 Construction of the Polynomial-Chaos based surrogate
As explained in Section 3.3.2, the interest of low-cost sensitivity analysis technique is twofold.
More than only identifying a ranking of main uncertainties, it can be used in order to build a
surrogate model (a Polynomial-Chaos based one in this case) on a reduced set of uncertainties,
i.e. the predominant ones. This is applied here to the temperature computed at a depth of 1.6
cm for different times. Note that this can be easily applied to a whatever quantity of interest,
but the surrogate will be not the same since the most important uncertainties can be different
with respect to the time, the depth and the quantity of interest.
In the case under consideration, first the uncertainties contributing the most to the variance
of the temperature at different times are computed (Figure 10 illustrates the ranking for a time
t= 80 s where the variance is maximal).
RR n°9175
16 Rivier, Lachaud, Congedo
(a) Recession (b) Char 2% location
(c) Virgin 98% location (d) Temperature, surface
Figure 6: Total variance divided by contributor, computed for several quantities, as a function
of time.
Secondly, the PC-based surrogate is constructed on a set of input parameters defined by the
predominant uncertainties. Here, uncertainties labeled as 24, 3, 4, 26, 1, 2, 9, 11 and 22 are
chosen as input parameters (See Section 2.2 for identifying each uncertainty).
Obviously, the reduction of the problem yields a loss of accuracy with respect to the statistics
computation, which could be estimated. Variance reduction is around 11%, which is mainly due
to the high number of significant parameters, which are neglected in the reduced problem.
Using the surrogate model, a whatever post-processing statistical analysis can be done for
free. As an example, the Probability Density Function (PDF) of the temperature at different
times are computed and represented in Figure 11. As it can be observed, gaussian-like PDF can
be observed.
One can see here an efficient method for studying TPS response under uncertainties.
5 Conclusion
The objective of the study was to propagate both material and pyrolysis gas composition un-
certainties on the thermal response of the TPS of a spacecraft during atmospheric entry. We
chose to study the entry of Stardust that was the first mission using the new generation of
Inria
Uncertainty propagation for heat and mass transfer in porous media 17
(a) Temperature, depth= 0.2 cm (b) Temperature, depth= 0.8 cm
(c) Temperature, depth= 1.6 cm (d) Temperature, bottom
Figure 7: Total variance divided by contributor, computed for several quantities, as a function
of time.
(a) Temperature, depth=1.6 cm (b) Temperature, depth = 2.4 cm
Figure 8: Total variance divided by contributor, computed for several quantities, as a function
of time. Here, uncertainties are classified into different groups.
RR n°9175
18 Rivier, Lachaud, Congedo
Figure 9: Hierarchy of the most relevant uncertain parameters to the temperature variance
(sorted: the higher, the bigger Sobol index) as a function of depth at t=80 s.
Figure 10: Sorted contributions at a depth of 1.6cm and at a time of t= 80 s
low-density carbon-phenolic ablators. Due to the high computational cost of varying pyrolysis
gas composition, a low-cost sensitivity analysis technique based on ANOVA has been used. To
clearly explicit the method in the field of material analysis, analytical derivations of the ANOVA
method were presented in the case of a well known deterministic heat transfer problem - tran-
sient conduction in a solid. Then, a sensitivity analysis technique based on anchored-ANOVA
was presented, permitting to treat problems described by expensive computed codes with several
uncertainties. This technique has been interfaced with PATO, a reactive porous material analysis
code distributed Open Source by NASA (https://software.nasa.gov/software/ARC-16680-1A, re-
trieve 26/06/2018). The suite of tools have been shown to be efficient to propagate uncertainties
and to successfully provide parameter hierarchies in the case of complicated simulations. The
clear contribution of uncertainties of pyrolysis gas composition has been revealed for the first
time. Perspectives of this work consist in using the suite of tool for inverse problems, e.g. use
optimization under uncertainties to optimize material properties for targeted conditions.
Inria
Uncertainty propagation for heat and mass transfer in porous media 19
Figure 11: PDF of the temperature at a depth of 1.6cm at different times.
References
[1] J. D. Anderson, Hypersonic and high temperature gas dynamics, Mac Graw-Hill, New-York,
1989.
[2] G. Duffa, Ablative Thermal Protection Systems Modeling, AIAA Education Series, 2013.
doi:10.2514/4.101717.
[3] M. Stackpoole, S. Sepka, I. Cozmuta, D. Kontinos, Post-flight evaluation of stardust sample
return capsule forebody heatshield material, AIAA paper 1202 (2008) 1–7.
[4] J. Lachaud, T. van Eekelen, J. B. Scoggins, T. E. Magin, N. N. Mansour, Detailed chemical
equilibrium model for porous ablative materials, International Journal of Heat and Mass
Transfer 90 (2015) 1034–1045, doi:10.1016/j.ijheatmasstransfer.2015.05.106.
[5] M. J. Wright, R. Beck, K. Edquist, D. Driver, S. Sepka, E. Slimko, W. Willcockson, Sizing
and margins assessment of the mars science laboratory aeroshell thermal protection system,
Journal of Spacecraft and Rockets 51 (4) (2014) 1125–1138.
[6] S. A. Sepka, M. J. Wright, Monte carlo approach to fiat uncertainties with applications for
mars science laboratory, Journal of Thermophysics and Heat Transfer 25 (2011) 516–522.
doi:10.2514/1.49804.
[7] S. A. Sepka, M. K. McGuire, J. V. Kam, A reliability comparison of classical and stochas-
tic thickness margin approaches for the orion heat shield resulting from material property
uncertainties, AIAA paper 1480. doi:10.2514/6.2018-1480.
RR n°9175
20 Rivier, Lachaud, Congedo
[8] H.-W. Wong, J. Peck, R. Bonomi, J. Assif, G. Reinisch, J. Lachaud, N. N. Mansour, Quan-
titative determination of species production from the pyrolysis of a phenol-formaldehyde
resin, Polymer Degradation and Stability 112 (2015) 122–131.
[9] B. K. Bessire, S. A. Lahankar, T. K. Minton, Pyrolysis of phenolic impregnated carbon
ablator (pica), Applied Materials and Interfaces 3 (7) (2015) 1383–1395.
[10] H.-W. Wong, J. Peck, J. Assif, F. Panerai, J. Lachaud, N. N. Mansour, Detailed analysis of
species production from the pyrolysis of phenolic impregnated carbon ablator., Journal of
Analytical and Applied Pyrolysis 122 (2016) 258–267.
[11] N. Fajraoui, M. Fahs, A. Younes, B. Sudret, Analyzing natural convection in porous en-
closure with polynomial chaos expansions: Effect of thermal dispersion, anisotropic perme-
ability and heterogeneity, International Journal of Heat and Mass Transfer 115 (2017) 205
– 224.
[12] X. Wang, C. Yan, S. Ju, Y. Zheng, J. Yu, Uncertainty analysis of laminar and turbulent
aeroheating predictions for mars entry, International Journal of Heat and Mass Transfer 112
(2017) 533 – 543.
[13] A. Turchi, P. M. Congedo, T. E. Magin, Thermochemical ablation modeling forward un-
certainty analysis-part i: Numerical methods and effect of model parameters, International
Journal of Thermal Sciences 118 (2017) 497 – 509.
[14] A. Turchi, P. M. Congedo, B. Helber, T. E. Magin, Thermochemical ablation modeling for-
ward uncertainty analysis-part ii: Application to plasma wind-tunnel testing, International
Journal of Thermal Sciences 118 (2017) 510 – 517.
[15] K. A. Trumble, I. Cozmuta, S. Sepka, P. Jenniskens, Post-flight aerothermal analysis of the
stardust sample return capsule, AIAA paper 1201 (2008) 1–15.
[16] J. Lachaud, A. Martin, T. van Eekelen, I. Cozmuta, Ablation test-case series #2, version
2.8, Jan. 2011, 8 p., prepared for the 5th Ablation Workshop, Feb. 28-March. 1, Lexington,
Kentucky.
[17] J. Lachaud, J. B. Scoggins, T. E. Magin, M. G. Meyer, N. N. Mansour, A generic local
thermal equilibrium model for porous reactive materials submitted to high temperatures,
International Journal of Heat and Mass Transfer 108 (2017) 1406–1417.
[18] H. Jasak, Error analysis and estimation for the finite volume method with applications to
fluid flows, Ph.D. thesis, Imperial College London (University of London) (1996).
[19] H. G. Weller, G. Tabor, H. Jasak, C. Fureby, A tensorial approach to computational con-
tinuum mechanics using object-oriented techniques, Computers in physics 12 (6) (1998)
620–631.
[20] J. Lachaud, N. N. Mansour, Porous material analysis toolbox based on openfoam and
applications, Journal of Thermophysics and Heat Transfer 28 (2) (2014) 191–202, doi:
10.2514/1.T4262.
[21] A. Omidy, F. Panerai, J. Lachaud, N. N. Mansour, I. Cozmuta, A. Martin, Code-to-code
comparison, and material response modeling of Stardust and MSL using PATO and FIAT,
NASA CR 218960 (2015) 1–36.
Inria
Uncertainty propagation for heat and mass transfer in porous media 21
[22] M. Mahzari, I. Cozmuta, I. Clark, R. Braun, An inverse parameter estimation methodology
for the analysis of aeroheating and thermal protection system experimental data, AIAA
paper 4027.
[23] G. E. B. Archer, A. Saltelli, I. M. Sobol’, Sensitivity measures, anova-like techniques and
the use of bootstrap, Journal of Statistical Computation and Simulation 58 (1997) 99–120.
[24] P. M. Congedo, C. Corre, J.-M. Martinez, Shape optimization of an airfoil in a bzt flow with
multiple-source uncertainties, Computer Methods in Applied Mechanics and Engineering
200 (1) (2011) 216–232.
[25] K. Tang, P. M. Congedo, R. Abgrall, Sensitivity analysis using anchored anova expansion
and high-order moments computation, International Journal for Numerical Methods in En-
gineering 102 (9) (2015) 1554–1584.
[26] X. Yang, M. Choi, G. Lin, G. Karniadakis, Adaptive {ANOVA} decomposition of stochastic
incompressible and compressible flows, Journal of Computational Physics 231 (4) (2012)
1587 – 1614.
[27] D. Xiu, G. E. Karniadakis, The Wiener–Askey Polynomial Chaos for Stochastic Differential
Equations, SIAM Journal on Scientific Computing 24 (2) (2002) 619–644.
[28] T. Crestaux, O. P. Le Maître, J. M. Martinez, Polynomial chaos expansion for sensitivity
analysis, Reliability engineering and system safety 94 (7) (2009) 1161 – 1172.
Appendix A
Detailed demonstrations of the analytical resolution of the transient heat transfer equation given
by
∂tT−α ∂2
xT= 0,(26)
where αis a constant diffusivity, appears to be hard to find in text books. We will present
its resolution using the Laplace transform for a semi-infinite unidimensional medium, with the
following boundary conditions
•T(∀x, t < 0) = T0
•T(x= 0, t >0) = Tw
corresponding to a homogeneous medium for which the surface temperature is suddenly raised.
Let us apply a variable change that will simplify the integration: θ(x, t) = T(x, t)−T0. Applying
the change of variable and the Laplace transform, Eq. 26 rewrites
Z∞
0
e−st∂2
xθdt −1
αZ∞
0
e−st∂tθdt = 0 (27)
In the Laplace space, the variable is defined as θ∗=R∞
0e−stθdt. After permutation (between
R∞
0·and ∂2
x·) and integration of the first integral, we obtain
∂2
xθ∗(x)−1
α[s θ∗(x)−θ(x, 0)] = 0 (28)
Thanks to the change of variable, we have θ(x, 0) = 0 and Eq. 28 simplifies into
∂2
xθ∗(x)−s
αθ∗(x)=0 (29)
RR n°9175
22 Rivier, Lachaud, Congedo
The solution of this second order ordinary differential equation is
θ∗(x) = Aexp(ps/α x) + Bexp(−ps/α x)(30)
where A and B are determined with the boundary conditions. On the semi-infinite domain, we
have
•A= 0 as the temperature has to have a limit when x tends towards infinity.
•θ∗(x= 0) = B. From θ(0, t) = Tw−T0, one obtains B= (Tw−T0)/s after applying
Laplace transform.
Hence, we have
θ∗(x) = Tw−T0
sexp(−ps/α x)(31)
Returning to the temporal space is done using the space change relation: exp(−a√s)/s ⇔
1−erf (a/(2√t), where erf is the error function. After some algebra and returning to variable
T, we obtain the physical temperature profile as a function of time
T(x, t) = T0+ (Tw−T0)1−erf x/√α
2√t (32)
Inria
RESEARCH CENTRE
BORDEAUX – SUD-OUEST
351, Cours de la Libération
Bâtiment A 29
33405 Talence Cedex
Publisher
Inria
Domaine de Voluceau - Rocquencourt
BP 105 - 78153 Le Chesnay Cedex
inria.fr
ISSN 0249-6399
