# Sensitivity Analysis of Nested Multiphysics Models Using Polynomial Chaos Expansions

Bruno Sudret, Thierry Yalamas, Emmanuel Noret, Pierre Willaume

► To cite this version:

Bruno Sudret, Thierry Yalamas, Emmanuel Noret, Pierre Willaume. Sensitivity Analysis of Nested Multiphysics Models Using Polynomial Chaos Expansions. 10th International Conference on Structural Safety and Reliability (ICOSSAR2009), Sep 2010, Osaka, Japan. hal-03674894

HAL Id: hal-03674894

https://hal.science/hal-03674894

Submitted on 21 May 2022

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Sensitivity Analysis of Nested Multiphysics Models Using Polynomial Chaos Expansions

Bruno Sudret<fnref n="1" />,<fnref n="2" />, Thierry Yalamas<fnref n="1" />, Emmanuel Noret<fnref n="1" />, Pierre Willaume<fnref n="1" />

Uncertainty quantification in computational mechanics has received much attention in the past few years. In this context, global sensitivity analysis (GSA) aims at quantifying which random input parameters (or combinations of parameters) are the most important in order to explain the variability of the model response. The present paper considers models of complex systems which are made of *nested* submodels, i.e. where input parameters of some submodels are obtained as the output of other submodels. A general framework is proposed in order to represent all the intermediate random variables (i.e. those variables that are obtained as response quantities of first-level submodels) by their polynomial chaos expansions. The coefficients of the various PC expansions are computed by regression and combined in order to obtain analytically the sensitivity (Sobol') indices. The crucial point is the representation of the joint PDF of intermediate input random variables, for which a Nataf transform is used. The approach is illustrated by sensitivity analysis of a multiscale micromechanics-based model of concrete elastic properties.

# 1 INTRODUCTION

Uncertainty quantification in computational mechanics has received much attention in the past few years. In this context, global sensitivity analysis (GSA) aims at quantifying which random input parameters (or combinations of parameters) are the most important in order to explain the variability of the model response. Various measures of importance have been proposed including partial correlation coefficients (PCC), standard regression coefficients (SRC) and Sobol' indices (Saltelli et al., 1999). The latter, which are usually computed using Monte Carlo simulation, allow the analyst to rank efficiently the input random variables whatever the complexity of the model.

In this paper we are interested in multiscale modelling in which several hierarchical models are nested, i.e. the response of one model corresponds to the input of another model and so on. This type of nested models may be represented by a network, in which each node is a model (having its own input and output vector) and some connections with the other nodes. They appear for instance when micromechanical models of materials at various scales are considered (i.e. a homogenized material property obtained at one scale is

used as the input parameter of a homogenization procedure at the next scale). The concept of nested models appears more generally in all multiphysics models. When considering nested models from the point of view of sensitivity analysis, it is desirable to establish importance measures (such as Sobol' indices) at various levels, i.e. the sensitivity of any response quantity at any level with respect to each variable of lower level (which may be itself the response of a lower level model).

Recently, sensitivity analysis based on Sobol' indices has been combined with polynomial chaos (PC) expansions, showing a great accuracy and computational efficiency (Sudret, 2006, 2008). PC expansions have been introduced in stochastic computational mechanics in the early 90's by Ghanem and Spanos (1991), based on the representation of stochastic processes due to Wiener (1938). Recently, non invasive computational techniques have allowed one to compute the coefficients of the PC series expansions from a set of evaluations of the deterministic model under consideration, namely projection (Ghiocel and Ghanem, 2002) or regression (Berveiller et al., 2006), see a review in Sudret (2007).

<footnote n="1">¹Phimeca Engineering, Centre d’Affaires du Zénith, 63800 Couron, France</footnote>

<footnote n="2">²LaMI, Institut Français de Mécanique Avancée et Université Blaise Pascal, Campus des Cézeaux, F-63175 Aubière Cedex, France</footnote>
In this paper, the technique developed in Sudret (2008) is extended to multiscale nested models. The difficulty lies in the representation of the intermediate random variables in a suitable manner to carry out a further PC expansion. The Nataf transform used in structural reliability is introduced for this purpose.

The basics of polynomial chaos expansions and global sensitivity analysis are recalled in Section 2 and 3 respectively. The adaptation of PC expansions for nested models is detailed in Section 4. The Nataf approximation is introduced. The application example is then presented in Section 5.

## 2 POLYNOMIAL CHAOS EXPANSIONS

### 2.1 Mathematical framework

Consider a mechanical system described by a numerical model $\mathcal{M}$ which can be analytical or more generally algorithmic (e.g. a finite element model). Suppose that this model has $\mathcal{M}$ uncertain input parameters which are represented by independent random variables $(X_1, \dots, X_M)$ gathered in a random vector $\mathbf{X}$ of prescribed joint probability density function $f_X(\mathbf{x})$. Hence the model response denoted by $Y = \mathcal{M}(\mathbf{X})$ is also random. For the sake of simplicity, $Y$ is assumed to be scalar throughout the paper (in case of a vector response $\mathbf{Y}$, the following derivations hold componentwise). Provided that the random variable $Y$ has a finite variance, it may be recast as follows (Soize and Ghanem, 2004):

$$Y = \mathcal{M}(\mathbf{X}) = \sum_{\alpha \in \mathbb{N}^M} a_{\alpha} \psi_{\alpha}(\mathbf{X}) \quad (1)$$

This expansion is referred to as the polynomial chaos (PC) representation of $Y$. The $a_\alpha$'s are unknown deterministic coefficients and the $\psi_\alpha$'s are multivariate polynomials which are orthonormal with respect to the joint PDF $f_X$ of the input random vector $\mathbf{X}$, i.e. $E[\psi_\alpha(\mathbf{X})\psi_\beta(\mathbf{X})] = 1$ if $\alpha = \beta$ and 0 otherwise. For instance, if $\mathbf{X}$ is a standard normal random vector, the $\psi_\alpha$ are normalized multivariate Hermite polynomials.

### 2.2 Estimation of the polynomial chaos coefficients

The PC coefficients can be estimated using a nonintrusive regression scheme (Berveiller et al., 2006; Sudret, 2007). This method requires the choice of a truncation of the PC i.e. a non empty finite set $\mathcal{A} = \{\alpha_0, \dots, \alpha_{P-1}\} \subset \mathbb{N}^M$ which contains the multi-indices of the retained basis polynomials $\psi_{\alpha_0}, \dots, \psi_{\alpha_{P-1}}$. The corresponding PC approximation is denoted by $Y_A \equiv \mathcal{M}_A(\mathbf{X}) = \sum_{\alpha \in \mathcal{A}} a_\alpha \psi_\alpha(\mathbf{X})$ which rewrites $Y_A = \mathbf{a}^\mathrm{T}\psi(\mathbf{X})$, by introducing the vector notation:

$$\mathbf{a} = \{a_{\alpha_0}, \dots, a_{\alpha_{P-1}}\}^\mathrm{T} \quad (2)$$

$$\psi(\mathbf{X}) = \{\psi_{\alpha_0}(\mathbf{X}), \dots, \psi_{\alpha_{P-1}}(\mathbf{X})\}^\mathrm{T} \quad (3)$$

Let us consider a set of realizations of $\mathbf{X}$ denoted by $\mathcal{X} = \{\mathbf{x}^{(1)}, \dots, \mathbf{x}^{(N)}\}$ and referred to as the *experimental design*. Let us denote by $\mathcal{Y}$ the associated set of model response, say $\mathcal{Y} = \{\mathcal{M}(\mathbf{x}^{(1)}), \dots, \mathcal{M}(\mathbf{x}^{(N)})\}$. The unknown coefficients $\mathbf{a}$ may be computed by performing a least-square minimization (Berveiller et al., 2006), i.e. by minimizing the mean-square truncation error $1/N \sum_{i=1}^N (\mathcal{M}(\mathbf{x}^{(i)}) - \mathcal{M}_A(\mathbf{x}^{(i)}))^2$. Using the above notation the solution reads:

$$\hat{\mathbf{a}} = (\Psi^\mathrm{T} \Psi)^{-1} \Psi^\mathrm{T} \mathcal{Y} \quad (4)$$

where $\Psi$ is a $N \times P$ matrix such that $\Psi_{ij} = \psi_{\alpha_j}(\mathbf{x}^{(i)}), i = 1, \dots, N, j = 0, \dots, P-1$. The size $N$ of the ED must be greater than $P$ to make this problem well posed.

### 2.3 Post-processing of the PC coefficients

Once the PC coefficients have been computed, it is straightforward to compute quantities of interest such as the probability density function and the statistical moments of the response quantity $Y$. Indeed, due to the orthonormality of the PC basis, the mean value of the model response is the first coefficient of the expansion, while the variance is obtained by the sum of the squares of the coefficients:

$$E[Y] = \hat{a}_0 \qquad \mathrm{Var}[Y] = \sum_{\alpha \in \mathcal{A} \setminus \{0\}} \hat{a}_\alpha^2 \quad (5)$$

In order to obtain a graphical representation of the response PDF, the series expansion may be simulated using Monte Carlo simulation. This yields a sample set of response quantities, say $\{y^{(i)}, i = 1, \dots, n\}$. From this set, an histogram may be built. Smoother representations may be obtained using *kernel smoothing* techniques, see e.g. Wand and Jones (1995):

$$\hat{f}_Y(y) = \frac{1}{n h_K} \sum_{i=1}^{n} K \left( \frac{y - y^{(i)}}{h_K} \right) \quad (6)$$

In this expression, $K(x)$ is a suitable positive function called *kernel*, and $h_K$ is the bandwidth parameter. Well-known kernels are the Gaussian kernel (which is the standard normal PDF) and the Epanechnikov kernel $K_E(x) = \frac{3}{4}(1 - x^2) \mathbf{1}_{|x| \le 1}$. The bandwidth parameter is selected according to the choice of the kernel and the sample size $n$. In the present case, the MC simulation of the PC expansion (which is analytical and polynomial) is rather inexpensive and usually negligible with respect to the model runs that were previously required to obtain the PC coefficients. Thus a large sample may be used for the kernel approximation, e.g. $n = 1,000,000$. In case of such large samples, the obtained kernel density is independent of the choice of the kernel function.
## 3 GLOBAL SENSITIVITY ANALYSIS

Global sensitivity analysis aims at quantifying which input parameters (or group of parameters) have a great influence on the response variability. Among various methods (see a review in Saltelli et al. (1999), variance decomposition (also known as ANOVA) methods have shown a great accuracy. The idea behind is to decompose the variance of the model response $Var[\mathcal{M}(\mathbf{X})]$ into contributions of each input parameter or combinations thereof. The Sobol' decomposition (Sobol', 1993) allows one to formulate the problem. Suppose the input parameters are defined on the M-dimensional unit cube $[0, 1]^M$ after some suitable transformation. The Sobol' decomposition of $\mathcal{M}(\mathbf{x})$ into summands of increasing dimension reads:

$$ \mathcal{M}(\mathbf{x}) = \mathcal{M}_0 + \sum_{i=1}^{M} \mathcal{M}_i(x_i) + \sum_{1 \le i < j \le M} \mathcal{M}_{ij}(x_i, x_j) + \dots + \mathcal{M}_{12\dots M}(\mathbf{x}) \quad (7) $$

where $\mathcal{M}_0$ is a constant and where it is imposed that the integral of each summand $\mathcal{M}_{i_1 \dots i_s}(x_{i_1}, \dots, x_{i_s})$ over any of its arguments is zero, i.e.:

$$ \int_{0}^{1} \mathcal{M}_{i_1 \dots i_s}(x_{i_1}, \dots, x_{i_s}) dx_{i_k} = 0 \quad \text{for } 1 \le k \le s \quad (8) $$

The classical properties of this decomposition are the following. The constant $\mathcal{M}_0$ is the mean value of the function: $\mathcal{M}_0 = \int_{[0,1]^M} \mathcal{M}(\mathbf{x}) d\mathbf{x}$ where $d\mathbf{x}$ stands for $dx_1 \dots dx_M$ for the sake of simplicity. Due to Eq.(8), the summands are orthogonal to each other in the following sense:

$$ \int_{[0,1]^M} \mathcal{M}_{i_1 \dots i_s}(x_{i_1}, \dots, x_{i_s}) \mathcal{M}_{j_1 \dots j_t}(x_{j_1}, \dots, x_{j_t}) d\mathbf{x} = 0 \quad \text{for } \{i_1, \dots, i_s\} \neq \{j_1, \dots, j_t\} \quad (9) $$

With the above assumptions, the decomposition in Eq.(7) is unique.

Consider now that the input parameters are independent random variables uniformly distributed over $[0, 1]$:

$$ \mathbf{X} = \{X_1, \dots, X_M\}^T, \quad X_i \sim \mathcal{U}(0, 1) \quad (10) $$

As a consequence, the model response $Y = \mathcal{M}(\mathbf{X})$ is a random variable, whose variance $D$ (also called total variance in the literature on global sensitivity) reads:

$$ D = \mathrm{Var}[\mathcal{M}(\mathbf{X})] = \int_{[0,1]^M} \mathcal{M}^2(\mathbf{x}) d\mathbf{x} - \mathcal{M}_0^2 \quad (11) $$

By integrating the square of Eq.(7) and by using (9), it is possible to decompose the total variance (11) as follows:

$$ D = \sum_{i=1}^{M} D_i + \sum_{1 \le i < j \le M} D_{ij} + \dots + D_{12\dots M} \quad (12) $$

where the partial variances appearing in the above expansion read:

$$ D_{i_1 \dots i_s} = \int_{[0,1]^s} \mathcal{M}^2_{i_1 \dots i_s}(x_{i_1}, \dots, x_{i_s}) dx_{i_1} \dots dx_{i_s} \quad (13) $$

The Sobol' indices are defined as follows:

$$ S_{i_1 \dots i_s} = D_{i_1 \dots i_s} / D \quad (14) $$

Each index $S_{i_1 \dots i_s}$ is a sensitivity measure describing which amount of the total variance is due to the uncertainties in the set of input parameters $\{i_1, \dots, i_s\}$. The first order indices $S_i$ give the influence of each parameter taken alone whereas the higher order indices account for possible mixed influence of various parameters.

## 4 PC EXPANSIONS AND NESTED MODELS

In many engineering problems, complex systems are modelled by a series of sub-models, which are linked to one another: the response quantities of some models are input to other submodels, thus the term “nested models” (see Figure 1).

### 4.1 Definition of nested models

Let us consider a complex model defined as a series of submodels $\{\mathcal{M}_q, q = 1, \dots, Q\}$. Each submodel has an input vector which can be split into two parts:

* a subvector denoted by $\mathbf{X}_q$, which corresponds to basic random variables for which a probabilistic model is available, say a joint probability density function $f_{\mathbf{X}_q}(\mathbf{x}_q)$.

* a subvector denoted by $\mathbf{Z}_q$, which corresponds to intermediate random variables, i.e. variables that are response quantities of other sub-model(s).

Let us denote by $\mathbf{Y}_q$ the response random vector of the q-th submodel. The following notation is used:

$$ \mathbf{Y}_q = \mathcal{M}_q(\mathbf{X}_q, \mathbf{Z}_q) \quad (15) $$

Note that the basic input random vectors $\{\mathbf{X}_q, \mathbf{X}_r, q \neq r\}$ are not necessarily independent (i.e. they may have some common components). Similarly, random vectors $\mathbf{X}_q, \mathbf{Z}_r$ may also be dependent, e.g. when $\mathbf{Z}_r$ is obtained as the response of a model $\mathcal{M}_r$ which also depends on $\mathbf{X}_q$ or a subvector of it.
## 4.2 PC expansion of the first-level model response
A so-called *first-level* submodel is a model $M_q(X_q)$ which only depends on a basic random vector $X_q$ with prescribed joint PDF $f_{X_q}(x_q)$. The input random vector is first approximated using the so-called *Nataf transform* (Nataf, 1962) whose definition is recalled in the next paragraph.

The Nataf transform denoted by $X_q = T(\xi)$ allows one to represent the input random vector in terms of independent standard normal variables. The transform is in general an approximation based on the margins and the linear correlation between the components. It is however exact when a) the components are independent, b) $X_q$ is a Gaussian vector or c) $X_q$ is defined by the set of margins and a Gaussian copula density (Nelsen (1999)), see Lebrun and Dutfoy (2009) for details).

In all cases, it is the model function $M_q$ composed with the Nataf transform $T$ that is represented by its PC expansion, namely:

$$Y_q = M_q(T(\xi)) \approx \sum_{\alpha \in A} a_{\alpha}^q \psi_{\alpha}(\xi) \quad (16)$$

## 4.3 Nataf transform

The Nataf transform (Nataf, 1962) has been introduced in structural reliability analysis in order to transform a random vector with prescribed marginals and correlation coefficients into a standard normal random vector, which is required e.g. in FORM analysis (Der Kiureghian and Liu, 1986; Ditlevsen and Madsen, 1996). The basic equations are now recalled.

* Suppose a random vector $X$ has prescribed marginal distributions, say $F_{X_i}(x_i), i = 1, \dots, M$ and correlation matrix $\rho$. Each component is first transformed into a standard normal variable $\xi_i, i = 1, \dots, M$:

$$\xi_i = \Phi^{-1}(F_{X_i}(x_i)) \quad (17)$$

* Then it is assumed that $\Xi = \{\xi_1, \dots, \xi_M\}^\mathrm{T}$ forms a standard normal correlated random vector whose joint PDF reads:

$$\begin{aligned} f_{\Xi}(\xi) &= \varphi_M(\xi; \tilde{\rho}) \\ &= (2\pi)^{-M/2} \exp \left[ -\frac{1}{2} \xi^\mathrm{T} \cdot \tilde{\rho}^{-1} \cdot \xi \right] \end{aligned} \quad (18)$$

The joint PDF of $X$ is thus approximated by the Nataf distribution:

$$f_X(x) = \frac{f_{X_1}(x_1) \dots f_{X_M}(x_M)}{\varphi(\xi_1) \dots \varphi(\xi_M)} \varphi_M(\xi, \tilde{\rho}) \quad (19)$$

The so-called pseudo-correlation matrix $\tilde{\rho}$ is computed componentwise by the consistency equation, which ensures that the correlation coefficient between any two components $\{X_i, X_j\}$ of $X$, as computed from (19) is the input correlation coefficient $\rho_{ij}$:

$$\rho_{ij} = \int_{\mathbb{R}^2} \left( \frac{x_i - \mu_{X_i}}{\sigma_{X_i}} \right) \left( \frac{x_j - \mu_{X_j}}{\sigma_{X_j}} \right) \varphi_2(\xi_i, \xi_j, \tilde{\rho}_{ij}) \, d\xi_i \, d\xi_j \quad (20)$$

Due to the burden of such a computation in the general case, values of $\tilde{\rho}_{ij}/\rho_{ij}$ have been tabulated for various couples of distributions $(f_{X_i}(x_i), f_{X_j}(x_j))$ (Der Kiureghian and Liu, 1986).

## 4.4 PC expansion of the next-level model response
Let us now consider a so-called *intermediate* submodel, i.e. a submodel $M_q(X_q, Z_q)$ which depends on the response of other submodels through the intermediate random vector $Z_q$.

As mentioned above, the joint PDF of $Z_q$ is not prescribed by the analyst, as it results from the propagation of some basic random vectors through other submodels. In the current framework, each intermediate random vector is thus represented as a set of polynomial chaos coefficients in a suitable PC basis.

In order to be able to cast the intermediate variables in a suitable way with respect to polynomial chaos expansions, the following strategy is adopted:

* the intermediate vectors, whose PC expansion is known, is sampled using crude Monte Carlo simulation or Latin Hypercube Sampling. Let us denote by $\mathcal{Z} \equiv \{z^{(i)}, i = 1, \dots, n\}$ the sample set and $\{z_j^{(i)}, i = 1, \dots, n\}$ the sample set of the j-th component of $Z_q$. The size of $\mathcal{Z}$ may be large (e.g. $10^5 - 10^6$) since the function to be simulated is analytical.

* For each component of $Z_q$, say $Z_{q,j}$, the empirical PDF and CDF is estimated by kernel smoothing techniques (Wand and Jones, 1995) from the corresponding sample set. For instance, using a Gaussian kernel function $\varphi(\cdot)$, the marginal PDF of $Z_{q,j}$ may be estimated by:

$$\hat{f}_{Z_{q,j}}(y) = \frac{1}{n h_K} \sum_{i=1}^{n} \varphi \left( \frac{y - z_j^{(i)}}{h_K} \right) \quad (21)$$

and the related CDF reads:

$$\hat{F}_{Z_{q,j}}(y) = \frac{1}{n} \sum_{i=1}^{n} \Phi \left( \frac{y - z_j^{(i)}}{h_K} \right) \quad (22)$$

where $\Phi(\cdot)$ is the standard normal CDF.
* The Pearson correlation matrix of the sample set $\mathcal{Z}$ is computed, say $\mathbf{R}_q$:

$$ \mathbf{R}_{q,k,l} = \frac{1}{n \hat{\sigma}_k \hat{\sigma}_l} \sum_{i=1}^{n} (z_k^{(i)} - \bar{z}_k) (z_l^{(i)} - \bar{z}_l) \quad (23) $$

where $\{\bar{z}_k, \bar{z}_l\}$ and $\{\hat{\sigma}_k, \hat{\sigma}_l\}$ are the empirical mean values and standard deviations of the samples of $Z_{q,k}$ and $Z_{q,l}$.

* The Nataf transform of vector $\mathbf{Z}_q$ into a standard normal random vector is computed numerically using the marginal CDF obtained by kernel smoothing (22) and the correlation matrix $\mathbf{R}_q$ (23). The pseudo-correlation matrix is computed numerically by solving Eq.(20).

# 5 APPLICATION EXAMPLE : HOMOGENIZATION OF CONCRETE PROPERTIES

In this section we aim at illustrating the above methodology on the modelling of the hydration of cement-based materials. From a deterministic point of view the hydration phenomena is modelled with a multiscale micromechanics-based approach develop by Bernard et al. (2003). The aim here is not to analyze the quantitative results on the example but to demonstrate a possible application of sensitivity analysis for nested models.

## 5.1 Micromechanics-hydration model

The modelling of the microstructure of concrete is broken up into four elementary levels (Figure 2). Two homogenization schemes are successively used: the self-consistent scheme for level II and the Mori-Tanaka scheme (Mori and Tanaka, 1973) for all the other levels.

**Level 0** The C-S-H matrix forms at early ages by hydration of $C_3S$ and $C_2S$. The characteristic length of this level varies between $10^{-8}$ and $10^{-6}$ m. At this level, at least two types of C-S-H exist with different mechanical properties and volume fractions.

### Model 1: Mori-Tanaka homogenization scheme

The input parameters of this model are the Young's modulus and Poisson's ratio of high density and low density C-S-H ($E_a, E_b, \nu_a, \nu_b$) and the high density C-S-H volume fraction denoted by $f_b$. The output of this model are the homogenized Young's modulus and Poisson's ratio of the C-S-H mix ($E_I, \nu_I$).

**Level I** The characteristic length of this level varies between $10^{-6}$ and $10^{-4}$ m. At this level, cement paste is composed from the C-S-H matrix with unhydrated cement products, large portlandite

crystals, aluminates and macro-porosity. Once the C-S-H phase percolates, the composite cement paste consists in inclusions embedded into a C-S-H matrix.

### Model 2: Self-consistent homogenization scheme

The input parameters of this second model are: Young's modulus and Poisson's ratio of hydrates (aluminates and CH) and their respective volume fraction, the volume fraction of water and void and the homogenized Young's modulus and Poisson's ratio resulting from Model #1 ($E_I, \nu_I$). The output of this model are the homogenized Young's modulus and Poisson's ratio of the cement paste ($E_{II}, \nu_{II}$).

**Level III** In this level, mortar is considered. It is made of a cement paste matrix, sand particle inclusions and an interfacial transition zone (ITZ). This ITZ is not taken into account in this study due to lack of data on its characteristics. The characteristic length of this level varies between $10^{-3}$ and $10^{-2}$ m.

### Model 3: Mori-Tanaka homogenization scheme

The input parameters of the third model are the Young's modulus and Poisson's ratio of sand and its volume fraction, and the homogenized Young's modulus and Poisson's ratio of the cement paste ($E_{II}, \nu_{II}$). The output of this model are the homogenized Young's modulus and Poisson's ratio of the mortar ($E_{III}, \nu_{III}$).

**Level IV** In this level, concrete, which is made of a mortar matrix, granular inclusions and an interfacial transition zone (ITZ), is considered. This ITZ is not taken into account in this study due to lack of data on its characteristics. The characteristic length of this level varies between $10^{-2}$ and $10^{-1}$ m.

### Model 4: Mori-Tanaka homogenization scheme

The input parameters of the fourth model are the Young's modulus and Poisson's ratio of aggregates and their volume fraction, and the homogenized Young's modulus and Poisson's ratio of the mortar ($E_{III}, \nu_{III}$). The output of this model are the homogenized Young's modulus and Poisson's ratio of concrete ($E_{IV}, \nu_{IV}$).

## 5.2 Probabilistic model

The probabilistic model is summarized in Table 3. Four independent lognormal random variables are introduced. The parameters of the distribution have been selected based on the values of ranges of variation given in Bernard et al. (2003). This probabilistic model may be certainly improved by discussions with
concrete experts. Nevertheless it is sufficiently accu-
rate to provide a significant application example.

5.3 Uncertainty results

Two types of information are obtained from this simu-
lation, namely the statistical moments of intermediate
and final output parameters, and global sensitivity re-
sults through Sobol' indices.

The statistical moments of the various interme-
diate quantities (Young's moduli and Poisson's ra-
tios of C-S-H mix, cement paste and mortar) and the
last-level output (Young's modulus and Poisson's ra-
tio of concrete) are given in Table 2. In the first two
columns, the moments are obtained from the "nested"
approach, in which each submodel is expanded as a
PC series and the coefficients are post-processed ac-
cording to Eq.(5). In the last two columns ("single
model"), the final output is considered as a unique
function of all the basic input parameters, and the
associated PC expansion is derived. This calculation
shows that the obtained mean and standard deviation
are equal to those obtained through the nested ap-
proach: from this point of view, the nested use of PC
expansions does not influence the quality of the last-
level results. The robustness of the nested approach is
thus validated by this comparison.

The Sobol' indices of the various intermediate and last-level output quantities with respect to their respective input parameters are gathered in Table 3. The very end product of the proposed methodology appears in this table: indeed, it is possible to calculate sensitivity indices of intermediate response quantities vs. intermediate variables, which could not be achieved by considering a single model.

The aim of this short example is not to give a
detailed interpretation of the results in terms of con-
crete formulations. Nevertheless, one can notice that
parameter E<sub>b</sub> (high density C-S-H Young's modulus)
has a strong influence on the final result. It may be
confirmed with a deeper analysis including a more
relevant probabilistic model.

6 CONCLUSION

The paper presents a framework for sensitivity anal-
ysis of complex models made of nested submodels.
The proposed methodology is based on polynomial
chaos expansions of the various submodels. The im-
portant step is the way intermediate random variables,
which are output variables of some given submodels,
may be transformed into standard variables in order to
expand the next-level model response quantities. The
Nataf transform is used for this purpose.

It is believed that the approximation introduced is
of second order in the context of sensitivity analysis.
However, this point should be further investigated. It
may be possible to use full generalized chaos expan-

sions that are consistent with a completely general
joint PDF of input parameters. However, this would
be far more complex since the basis of expansion is
no more polynomial.(Soize and Ghanem, 2004; Su-
dret, 2007).

Furthermore, the interpretation of the Sobol' indices computed by this approach may be also carried out carefully: for the intermediate models, the input variables are possibly correlated, and then transform through the Nataf approximation into standard normal variables. Rigorously speaking, the indices computed as the post-processing of the chaos coefficients refer to those standard normal variables and not directly to the underlying physical variables. A rigourous interpretation would require the study of sensitivity indices with respect to correlated variables, which is somehow an open question, although some approaches have been recently proposed (Xu and Gertner, 2008).

REFERENCES

Bernard, O., F.-J. Ulm, and E. Lemarchand (2003). A multiscale micromechanics-hydration model for the early-age elastic properties of cement-based materials. *Cement Conc. Res.* 33(9).

Berveiller, M., B. Sudret, and M. Lemaire (2006). Stochastic finite elements: a non intrusive approach by regression. *Eur. J. Comput. Mech.* 15(1-3), 81–92.

Der Kiureghian, A. and P. Liu (1986). Structural reliability under incomplete probability information. *J. Eng. Mech.* 112(1), 85–104.

Ditlevsen, O. and H. Madsen (1996). *Structural reliability methods*. J. Wiley and Sons, Chichester.

Ghanem, R. and P. Spanos (1991). *Stochastic finite elements – A spectral approach*. Springer Verlag. (Reedited by Dover Publications, 2003).

Ghiocel, D. and R. Ghanem (2002). Stochastic finite element analysis of seismic soil-structure interaction. *J. Eng. Mech.* 128, 66–77.

Lebrun, R. and A. Dutfoy (2009). An innovating analysis of the Nataf transformation from the copula viewpoint. *Prob. Eng. Mech.* Accepted for publication.

Mori, T. and K. Tanaka (1973). Average stress in matrix and average elastic energy of materials with misfitting inclusion. *Acta Metallurgica* 21(5), 1605–1609.

Nataf, A. (1962). Détermination des distributions
dont les marges sont données. *C. R. Acad. Sci.
Paris* 225, 42–43.
Nelsen, R. (1999). *An introduction to copulas*, Volume 139 of *Lecture Notes in Statistics*. Springer.

Saltelli, A., S. Tarantola, and K. Chan (1999). A quantitative, model independent method for global sensitivity analysis of model output. *Technometrics* 41(1), 39–56.

Sobol', I. (1993). Sensitivity estimates for nonlinear mathematical models. *Math. Modeling & Comp. Exp.* 1, 407–414.

Soize, C. and R. Ghanem (2004). Physical systems with random uncertainties: chaos representations with arbitrary probability measure. *SIAM J. Sci. Comput.* 26(2), 395–410.

Sudret, B. (2006). Global sensitivity analysis using polynomial chaos expansions. In P. Spanos and G. Deodatis (Eds.), *Proc. 5th Int. Conf. on Comp. Stoch. Mech (CSM5)*, Rhodos, Greece.

Sudret, B. (2007). *Uncertainty propagation and sensitivity analysis in mechanical models – Contributions to structural reliability and stochastic spectral methods*. Habilitation à diriger des recherches, Université Blaise Pascal, Clermont-Ferrand, France.

Sudret, B. (2008). Global sensitivity analysis using polynomial chaos expansions. *Reliab. Eng. Sys. Safety*. In press.

Wand, M. and M. Jones (1995). *Kernel smoothing*. Chapman and Hall.

Wiener, N. (1938). The homogeneous chaos. *Amer. J. Math.* 60, 897–936.

Xu, C. and G. Gertner (2008). Uncertainty and sensitivity analysis for models with correlated parameters. *Reliab. Eng. Sys. Safety*, 1563–1573.

<figure><img src="image_2.png" /><figcaption>Figure 1: Decomposition of a complex model into submodels</figcaption></figure>
<figure><img src="image_3.png" /><figcaption>Figure 2: Multiscale microstructure of cement-based material (Bernard et al., 2003)</figcaption></figure>

<table><caption>Table 1: Probabilistic description of the input parameters</caption><thead><tr><th rowspan="2">Level</th><th rowspan="2">Parameter description</th><th rowspan="2">Name</th><th colspan="3">Material properties</th><th colspan="3">Probabilistic model</th></tr><tr><th>Mean</th><th>Min</th><th>Max</th><th>Type of PDF</th><th>Mean</th><th>St. Dev.</th></tr></thead><tbody><tr><td rowspan="5">0</td><td>Low density C-S-H Young's Modulus</td><td>E<sub>a</sub> (GPa)</td><td>21.7</td><td>19.5</td><td>23.9</td><td>Lognormal</td><td>21.7</td><td>0.7</td></tr><tr><td>High density C-S-H Young's Modulus</td><td>E<sub>b</sub> (GPa)</td><td>29.4</td><td>27</td><td>31.8</td><td>Lognormal</td><td>29.4</td><td>0.7</td></tr><tr><td>Low density C-S-H Poisson's ratio</td><td>ν<sub>a</sub></td><td>0.24</td><td></td><td></td><td>Fixed</td><td></td><td></td></tr><tr><td>High density C-S-H Poisson's ratio</td><td>ν<sub>b</sub></td><td>0.24</td><td></td><td></td><td>Fixed</td><td></td><td></td></tr><tr><td>High density C-S-H volume fraction</td><td>f<sub>b</sub></td><td>0.3</td><td></td><td></td><td>Fixed</td><td></td><td></td></tr><tr><td rowspan="10">I</td><td>CH hydrate Young's Modulus</td><td>E<sub>1</sub> (GPa)</td><td>38</td><td>33</td><td>43</td><td>Lognormal</td><td>38</td><td>1.5</td></tr><tr><td>CH hydrate Poisson's ratio</td><td>ν<sub>1</sub></td><td>0.305</td><td>0.305</td><td>0.325</td><td>Lognormal</td><td>0.315</td><td>0.001</td></tr><tr><td>CH volume fraction</td><td>f<sub>1</sub></td><td>0.156</td><td></td><td></td><td>Fixed</td><td></td><td></td></tr><tr><td>Aluminate hydrate Young's Modulus</td><td></td><td colspan="3">E<sub>I</sub> (from level-I homogenization)</td><td></td><td></td><td></td></tr><tr><td>Aluminate hydrate Poisson's ratio</td><td></td><td colspan="3">ν<sub>I</sub> (from level-I homogenization)</td><td></td><td></td><td></td></tr><tr><td>Aluminate hydrate volume fraction</td><td>f<sub>2</sub></td><td>0.169</td><td></td><td></td><td>Fixed</td><td></td><td></td></tr><tr><td>Water volume fraction</td><td>f<sub>3</sub></td><td>0.062</td><td></td><td></td><td>Fixed</td><td></td><td></td></tr><tr><td>Void volume fraction</td><td>f<sub>4</sub></td><td>0.075</td><td></td><td></td><td>Fixed</td><td></td><td></td></tr><tr><td>Sand Young's Modulus</td><td>E<sub>s</sub> (GPa)</td><td>35</td><td>30</td><td>40</td><td>Fixed</td><td></td><td></td></tr><tr><td>Sand Poisson's ratio</td><td>ν<sub>s</sub></td><td>0.3</td><td></td><td></td><td>Fixed</td><td></td><td></td></tr><tr><td>Sand volume fraction</td><td>f<sub>s</sub></td><td>0.1</td><td></td><td></td><td>Fixed</td><td></td><td></td></tr><tr><td rowspan="5">II</td><td>Cement paste Young's Modulus</td><td></td><td colspan="3">E<sub>II</sub> (from level-II homogenization)</td><td></td><td></td><td></td></tr><tr><td>Cement paste Poisson's ratio</td><td></td><td colspan="3">ν<sub>II</sub> (from level-II homogenization)</td><td></td><td></td><td></td></tr><tr><td>Aggregates Young's Modulus</td><td>E<sub>g</sub> (GPa)</td><td>35</td><td>30</td><td>40</td><td>Fixed</td><td></td><td></td></tr><tr><td>Aggregates Poisson's ratio</td><td>ν<sub>g</sub></td><td>0.3</td><td></td><td></td><td>Fixed</td><td></td><td></td></tr><tr><td>Aggregates volume fraction</td><td>f<sub>g</sub></td><td>0.1</td><td></td><td></td><td>Fixed</td><td></td><td></td></tr><tr><td rowspan="4">III</td><td>Mortar Young's Modulus</td><td></td><td colspan="3">E<sub>III</sub> (from level-III homogenization)</td><td></td><td></td><td></td></tr><tr><td>Mortar Poisson's ratio</td><td></td><td colspan="3">ν<sub>III</sub> (from level-III homogenization)</td><td></td><td></td><td></td></tr></tbody></table>
<table><caption>Table 2: Mean value and standard deviation of the elastic properties resulting from each step of homogenization: single model vs. nested models</caption><thead><tr><th rowspan="2"></th><th colspan="2">Nested models</th><th colspan="2">Single model</th></tr><tr><th>Mean</th><th>Std. dev.</th><th>Mean</th><th>Std. dev.</th></tr></thead><tbody><tr><td>E<sub>I</sub></td><td>35.517</td><td>1.037</td><td></td><td></td></tr><tr><td>ν<sub>I</sub></td><td>0.111</td><td>0.011</td><td></td><td></td></tr><tr><td>E<sub>II</sub></td><td>26.548</td><td>0.657</td><td></td><td></td></tr><tr><td>ν<sub>II</sub></td><td>0.169</td><td>0.006</td><td></td><td></td></tr><tr><td>E<sub>III</sub></td><td>33.975</td><td>0.474</td><td></td><td></td></tr><tr><td>ν<sub>III</sub></td><td>0.111</td><td>0.007</td><td></td><td></td></tr><tr><td>E<sub>IV</sub></td><td>34.947</td><td>0.152</td><td>34.949</td><td>0.152</td></tr><tr><td>ν<sub>IV</sub></td><td>0.132</td><td>0.005</td><td>0.132</td><td>0.005</td></tr></tbody></table>

<table><caption>Table 3: Sobol' indices of the various levels</caption><thead><tr><th>Level</th><th>Index definition</th><th>Value</th></tr></thead><tbody><tr><td rowspan="4">I</td><td>E<sub>I</sub>/E<sub>a</sub></td><td>0.02</td></tr><tr><td>E<sub>I</sub>/E<sub>b</sub></td><td>0.98</td></tr><tr><td>ν<sub>I</sub>/E<sub>a</sub></td><td>0.43</td></tr><tr><td>ν<sub>I</sub>/E<sub>b</sub></td><td>0.57</td></tr><tr><td rowspan="8">II</td><td>E<sub>II</sub>/E<sub>I</sub></td><td>0.97</td></tr><tr><td>E<sub>II</sub>/ν<sub>I</sub></td><td>0.00</td></tr><tr><td>E<sub>II</sub>/E<sub>1</sub></td><td>0.03</td></tr><tr><td>E<sub>II</sub>/ν<sub>1</sub></td><td>0.00</td></tr><tr><td>ν<sub>II</sub>/E<sub>I</sub></td><td>0.06</td></tr><tr><td>ν<sub>II</sub>/ν<sub>I</sub></td><td>0.26</td></tr><tr><td>ν<sub>II</sub>/E<sub>1</sub></td><td>0.00</td></tr><tr><td>ν<sub>II</sub>/ν<sub>1</sub></td><td>0.68</td></tr><tr><td rowspan="4">III</td><td>E<sub>III</sub>/E<sub>II</sub></td><td>1.00</td></tr><tr><td>E<sub>III</sub>/ν<sub>II</sub></td><td>0.00</td></tr><tr><td>ν<sub>III</sub>/E<sub>II</sub></td><td>0.85</td></tr><tr><td>ν<sub>III</sub>/ν<sub>II</sub></td><td>0.15</td></tr><tr><td rowspan="4">IV</td><td>E<sub>IV</sub>/E<sub>III</sub></td><td>0.54</td></tr><tr><td>E<sub>IV</sub>/ν<sub>III</sub></td><td>0.46</td></tr><tr><td>ν<sub>IV</sub>/E<sub>III</sub></td><td>0.88</td></tr><tr><td>ν<sub>IV</sub>/ν<sub>III</sub></td><td>0.12</td></tr></tbody></table>