Probabilistic analysis of concrete cracking using stochastic finite element methods: application to nuclear containment buildings at early age

Abstract

In the case of quasi-homogeneously applied tensile loads, the intrinsic scattering of concrete properties leads to spatially random strain localization, crack initiation and propagation. The modelling of such spatial randomness, in the case of Equivalent-Homogeneous-Material Finite Elements based approaches, can be achieved thanks to the use of Random Fields. However, when aiming at probabilistic analyses, numerous realizations are required which induces a hefty computational time and restricts their applicability to the modelling of large concrete structures. In this contribution, an original probabilistic coupling strategy is provided based on non-intrusive Stochastic Finite Elements Methods. It consists of defining an explicit Surface Response of the cracking patterns expressed in terms of the most influential inputs using an Adaptive Surface Response Method combined to a Polynomial Chaos Expansion Method. Direct Monte Carlo Method is then applied—to the explicit Surface Response of the cracking patterns—to achieve Global Sensitivity Analysis, Uncertainties Quantification and probabilistic modelling at a reasonable cost. The defined strategy is validated based on a Representative Structural Volume of a 1:3 scaled experimental Containment Building at early age using a weakly coupled thermo-mechanical model. As a result, the study quantifies the effect of the most influential parameters (the Young’s modulus—the tensile strength—the coefficients of thermal expansion and autogenous shrinkages) on concrete cracking at early age and provides accurate numerical prediction of the cracking patterns (cracks’ number, opening and spacing values) observed on site and their frequencies.

Appendix: Brief reminder about polynomial chaos expansion methods (PCE)

1.1 Appendix 1: PCE

Let’s consider ${\mathbf{\mathcal{M}}}$ a computational model whose input parameters are represented by a random vector ${\mathbf{X}} = \left\{ {\varvec{X}_{{1 \le \varvec{i} \le \varvec{N}}} } \right\}$. The model’s random outputs ${\mathbf{Y}}$ verify ${\mathbf{Y}} = \left\{ {\varvec{Y}_{{1 \le \varvec{j} \le \varvec{N^{\prime}}}} } \right\} = {\mathbf{\mathcal{M}}}\left( {\mathbf{X}} \right)$. Given the physical nature of our problem, ${\mathbf{Y}}$ is assumed to have a finite variance and can be fully defined using the following Hilbertian representation [22]:

$$ {\mathbf{Y}} = {\mathbf{\mathcal{M}}}\left( {\mathbf{X}} \right) = \mathop \sum \limits_{{{\mathbf{k}} = 0}}^{ + \infty } y_{k}\Psi _{k} \left( {\mathbf{X}} \right) $$

(14)

where $\left\{ {\Psi _{{k \in {\mathbb{N}}}} } \right\}$ a numerable set of random variables forming the Hilbertian basis and $\left\{ {y_{{k \in {\mathbb{N}}}} } \right\}$ are coordinates within that basis.

Several choices are possible for $\left\{ {\Psi _{{k \in {\mathbb{N}}}} } \right\}$ verifying Eq. 14. Herein, a particular focus is granted to PCE in which the basis terms are multivariate orthonormal polynomials [23].The selection of the polynomial basis $\left\{ {P_{{i \in {\mathbb{N}}}} } \right\}$ to be associated with each random parameter $X_{i}$ depends on its marginal PDF $f_{i} \left( {x_{i} } \right)$. For instance, and under the hypothesis of independent random variables $\left\{ {X_{i} } \right\}$, standard normal distributions $X_{i} \sim\aleph \left( {0,1} \right)$ lead to the use of Hermite polynomials $\left\{ {{\text{He}}_{i} } \right\}$ and uniform distributions $X_{i} \sim {U}\left( { - 1,1} \right)$ lead to the use of Legendre polynomials $\left\{ {{\text{Le}}_{i} } \right\}$:

$$ \begin{aligned} X_{i} \sim \aleph \left( {0,1} \right) & \to \left\{ {\begin{array}{*{20}l} {\left\langle {P_{m} ,P_{n} } \right\rangle_{i} = \int\limits_{{D_{{X_{i} }} }} {P_{m} \left( x \right)P_{n} \left( x \right)f_{{X_{i} }} \left( x \right)\;{\text{d}}x = \delta_{mn} } } \hfill \\ {\left\{ {P_{{k \in {\mathbb{N}}}} } \right\} = \left\{ {\frac{{{\text{He}}_{k} }}{{\sqrt {k!} }}} \right\}} \hfill \\ {\left\{ {\begin{array}{*{20}l} {{\text{He}}_{ - 1} \left( x \right) = {\text{He}}_{0} \left( x \right) = 1} \hfill \\ {{\text{He}}_{k \ge 1} \left( x \right) = x {\text{He}}_{k - 1} \left( x \right) - \left( {k - 1} \right) {\text{He}}_{k - 2} \left( x \right)} \hfill \\ \end{array} } \right.} \hfill \\ \end{array} } \right. \\ & \to \left\{ {\begin{array}{*{20}l} {P_{0} \left( x \right) = 1} \hfill \\ {P_{1} \left( x \right) = x} \hfill \\ {P_{2} \left( x \right) = \frac{{x^{2} - 1}}{\sqrt 2 }} \hfill \\ {P_{3} \left( x \right) = \frac{{x\left( {x^{2} - 3} \right)}}{\sqrt 6 }} \hfill \\ \end{array} } \right. \\ X_{i} \sim U\left( { - 1,1} \right) & \to \left\{ {\begin{array}{*{20}l} {\left\langle {P_{m} ,P_{n} } \right\rangle_{i} = \int\limits_{{D_{{X_{i} }} }} {P_{m} \left( x \right)P_{n} \left( x \right)f_{{X_{i} }} \left( x \right)\;{\text{d}}x = \delta_{mn} } } \hfill \\ {\left\{ {P_{{k \in {\mathbb{N}}}} } \right\} = \left\{ {\frac{{{\text{Le}}_{k} }}{{\sqrt {\frac{1}{2k + 1}} }}} \right\}} \hfill \\ {{\text{Le}}_{k} = 2^{k} \mathop \sum \limits_{i = 0}^{k} x^{i} \left( {\begin{array}{*{20}c} k \\ i \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {\frac{k + i - 1}{2}} \\ k \\ \end{array} } \right)} \hfill \\ \end{array} } \right. \\ & \to \left\{ {\begin{array}{*{20}l} {P_{0} \left( x \right) = 1} \hfill \\ {P_{1} \left( x \right) = \sqrt 3 x} \hfill \\ {P_{2} \left( x \right) = \frac{\sqrt 5 }{2}\left( {3x^{2} - 1} \right)} \hfill \\ {P_{3} \left( x \right) = \frac{\sqrt 7 }{2} \times \left( {5x^{2} - 3} \right)} \hfill \\ \end{array} } \right. \\ \end{aligned} $$

(15)

where $\delta_{mn}$ the Kronecker symbol equal to 1 when n = m and 0 otherwise, $D_{{X_{i} }}$ the support or definition domain of $X_{i}$.

As the random vector ${\mathbf{X}}$ contains N random variables, a multivariate polynomial construction is needed to express ${\mathbf{Y}}$ as shown in Eq. 14. It can be built up simply by product tensorization as following [22]:

$$ \psi_{{\alpha = \left\{ {\alpha_{1 \le i \le N} } \right\}}} \left( {\mathbf{x}} \right) = \mathop \prod \limits_{i = 1}^{N} P_{{\alpha_{i} }} \left( {x_{i} } \right) $$

(16)

where $\left\{ {\psi_{{\alpha = \left\{ {\alpha_{{1 \le i \le {\text{N}}}} } \right\}}} ;\;\alpha_{i} \in {\mathbb{N}}} \right\}$ a countable orthonormal basis to represent the random response ${\mathbf{Y}}$.$\alpha = \left\{ {\alpha_{1 \le i \le N} } \right\}$ a global index referring to the order of each polynomial $\alpha_{i}$ associated with the parameter’s realization $x_{i}$.

Equation 14 is an infinite sum. In practice, only a finite number of terms is to be retained for the PCE (truncated expression). Such number can be defined in a way so as not to exceed a certain polynomial degree $Q \le \mathop \sum \nolimits_{{{\mathbf{i}} = 1}}^{\varvec{N}} q_{i}$ with $q_{i}$ the highest polynomial degree of the polynomial basis $\left\{ {P_{{0 \le i \le q_{i} }} } \right\}$ associated with each random input $X_{i}$:

$$ {\mathbf{Y}} \approx \widehat{{\mathbf{Y}}} = \mathop \sum \limits_{{\sum \alpha_{i} = 0}}^{\varvec{Q}} y_{\alpha }\Psi _{\alpha } \left( {\mathbf{X}} \right) $$

(17)

Consequently, the number of terms in Eq. 17 should be $\left( {\begin{array}{*{20}c} {N + Q} \\ Q \\ \end{array} } \right) = \frac{{\left( {N + Q} \right)!}}{N!Q!}$ which increases polynomially with N and Q (curse of dimensionality [39]). As the required order $Q$ for an accurate model approximation is not known a priori, conventional values around ~ 3–5 are considered in practice [11].

1.2 Appendix 2: PCE coefficients identification

The identification of $y_{\alpha }$ coefficients in Eq. 17 requires several calls of the model. In the case of regression approaches [40], the number of calls should be higher than the number of unknowns for the problem to be well-posed: $N_{\text{call}} \ge \frac{{\left( {N + Q} \right)!}}{N!Q!}$. This however, does not take advantage of the orthonormality rule of the basis $\left\{ {\psi_{{\alpha = \left\{ {\alpha_{1 \le i \le N} } \right\}}} ;\;\alpha_{i} \in {\mathbb{N}}} \right\}$ unlike projection methods where each coordinate $y_{\alpha }$ writes:

$$ y_{\alpha } = \int\limits_{{D_{X} }} {\Psi _{\alpha } \left( {\mathbf{x}} \right)f_{X} \left( {\mathbf{x}} \right){\mathbf{\mathcal{M}}}\left( {\mathbf{x}} \right)\;{\text{d}}{\mathbf{x}}} $$

(18)

with $f_{X} \left( {\mathbf{x}} \right) = \mathop \prod \nolimits_{i = 1}^{N} f_{{X_{i} }} \left( {x_{i} } \right)$ the joint PDF of the random vector ${\mathbf{X}}$. The problem then is reduced to the evaluation of the previous integration using classical integration methods, in particular the Gauss-Quadrature Method:

$$ y_{\alpha } \approx \mathop \sum \limits_{{j_{1} = 1}}^{{{\text{S}}_{1} }} \ldots \mathop \sum \limits_{{j_{N} = 1}}^{{S_{N} }} w_{{j_{1} }}^{1} \ldots w_{{j_{N} }}^{N} \Psi _{{\left\{ {\alpha_{1} , \ldots ,\alpha_{N} } \right\}}} \left( {{\mathbf{x}}_{{1,{\mathbf{j}}_{1} }} , \ldots , {\mathbf{x}}_{{{\mathbf{N}},{\mathbf{j}}_{{\mathbf{N}}} }} } \right) {\mathbf{\mathcal{M}}}\left( {{\mathbf{x}}_{{1,{\mathbf{j}}_{1} }} , \ldots , {\mathbf{x}}_{{{\mathbf{N}},{\mathbf{j}}_{{\mathbf{N}}} }} } \right) $$

(19)

with $\left( {x_{{k,j_{k} }} ;\;w_{{j_{k} }}^{k} } \right)$ the integration points and the associated weights with respect to the distribution $f_{{X_{k} }}$. Compared to simulation methods where a relatively high number of random integration points is chosen (MCM or QMCM [41, 42]), quadrature-based approaches [43, 44] are preferred herein to limit the computational time. In that sense, it is worth recalling that, in the case of isotropic formulae, the Gauss-Quadrature Method allows to integrate any polynomial function of a degree $2n - 1$ with $n$ suitable integration points. Consequently, for the PCE of an order $Q$ and a polynomial basis with a maximum order $q$, the integrand in Eq. 18 has a maximal degree of $Qq$ requiring $Qq + 1$ integration points per parameter and $\left( {Qq + 1} \right)^{N}$ calls of the model—if a full grid is retained. For high number of inputs (more than ten inputs) such curse of dimensionality can be reduced by using adapted levels within the Smolyak quadrature scheme [45] or adaptive PCE methods [46].However such level is not known a priori and is defined based on a user-defined error threshold (only known if sufficient feedbacks are available for the problem of interest).

Herein, the default Gauss-Quadrature Method is used. One can notice that there are two ways to truncate the PCE:

• Option 1 Fixing the polynomial degrees for each parameter to a fixed value $q$. This would lead to a global degree of $Q = Nq$ in Eq. 17 (isotropic formulae). Using the Gauss-Quadrature Method, $\left( {Nq^{2} + 1} \right)^{N}$ calls of the model need to be performed.

• Option 2 Fixing the global polynomial degree to a fixed value $Q = q$. The number of model calls is hugely reduced to $\left( {q + 1} \right)^{N}$ as all terms $y_{\alpha }$ of which $\Psi _{\alpha }$ verifies $\sum \alpha_{i} > Q$ are discarded.

Clearly, option 2 is less expensive than option 1 and is, therefore, retained herein. One is also reminded that, beyond accuracy issues, the higher $q$ is the wider the covered domain of each random input $X_{i}$ is (by definition a Surface Response is only valid within its estimation domain). This aspect should be considered primarily (whether the scheme is adaptive or not) when selecting the $q$ value to avoid extrapolation errors within the physically admissible variation domains.

1.3 Appendix 3: Variance-based sensitivity analysis

The influence of each input on the computed response can be estimated using Global Sensitivity Analysis techniques. Particularly, Sobol’ indexes [12] allow the decomposition of the output’s variance into fractions that can be attributed to each input or set of inputs. Again, exploring the orthonormality rule, Sobol’ indexes might be easily derived [47] after a convenient reordering of the PCE terms:

$$ \begin{aligned} \widehat{{\mathbf{Y}}}\left( {\mathbf{x}} \right) & = \mathop \sum \limits_{{\sum \alpha_{i} = 0}}^{\varvec{Q}} y_{\alpha }\Psi _{\alpha } \left( {\mathbf{x}} \right) \\ & = \varvec{y}_{0} + \sum\limits_{{1 \le {\mathbf{i}}_{1} \le {\mathbf{N}}}} {\sum\limits_{{{\varvec{\upalpha}} \in {\varvec{\Gamma}}_{{{\mathbf{i}}_{1} }} }} {y_{\alpha }\Psi _{\alpha } \left( {{\mathbf{x}}_{{{\mathbf{i}}_{1} }} } \right) + \sum\limits_{{1 \le {\mathbf{i}}_{1} < {\mathbf{i}}_{2} \le {\mathbf{N}}}} {\sum\limits_{{{\varvec{\upalpha}} \in {\varvec{\Gamma}}_{{{\mathbf{i}}_{1} ,\varvec{i}_{2} }} }} {y_{\alpha }\Psi _{\alpha } \left( {{\mathbf{x}}_{{{\mathbf{i}}_{1} }} ,{\mathbf{x}}_{{{\mathbf{i}}_{2} }} } \right) \; + \cdots } } } } \\ & \quad + \,\sum\limits_{{1 \le {\mathbf{i}}_{1} < \cdots < {\mathbf{i}}_{{\mathbf{s}}} \le {\mathbf{N}}}} {\sum\limits_{{{\varvec{\upalpha}} \in {\varvec{\Gamma}}_{{{\mathbf{i}}_{1} , \ldots ,\varvec{i}_{\varvec{s}} }} }} {y_{\alpha }\Psi _{\alpha } \left( {{\mathbf{x}}_{{{\mathbf{i}}_{1} }} , \ldots ,{\mathbf{x}}_{{{\mathbf{i}}_{{\mathbf{s}}} }} } \right) \; + \cdots } } \\ & \quad +\, \sum\limits_{{{\varvec{\upalpha}} \in {\varvec{\Gamma}}_{{1, \ldots ,\varvec{N}}} }} {y_{\alpha }\Psi _{\alpha } \left( {{\mathbf{x}}_{1} , \ldots ,{\mathbf{x}}_{{\mathbf{N}}} } \right) } \\ \end{aligned} $$

(20)

with $y_{0} = E\left[ {\widehat{{\mathbf{Y}}}} \right]$ the mean estimate of $\widehat{{\mathbf{Y}}}$, $\Gamma _{{i_{1} , \ldots ,i_{s} }} = \left\{ {\alpha : \begin{array}{*{20}c} {\alpha_{k} > 0\quad k \in \left( {i_{1} , \ldots ,i_{s} } \right)} \\ {\alpha_{k} = 0 \quad k \notin \left( {i_{1} , \ldots ,i_{s} } \right)} \\ \end{array} } \right\}$ the set of ${\alpha }$ multi-indexes such that only the indexes $\left( {i_{1} , \ldots ,i_{s} } \right)$ are non-null.

The uniqueness of decomposition in Eq. 20 leads to the conclusion that it is exactly the Sobol’ decomposition of $\widehat{Y}$. This means that the s-ith Sobol’ indexes are straightforward:

$$ I_{{i_{1} , \ldots ,i_{s} }}^{\text{PC}} = \frac{{\sum\nolimits_{{{\varvec{\upalpha}} \in {\varvec{\Gamma}}_{{{\mathbf{i}}_{1} , \ldots ,\varvec{i}_{\varvec{s}} }} }} {\left( {y_{\alpha } } \right)^{2} } }}{{\mathop \sum \nolimits_{{\sum \alpha_{i} = 1}}^{\varvec{Q}} \left( {y_{\alpha } } \right)^{2} }} $$

(21)

where ${\text{VAR}}\left[ {\widehat{{\mathbf{Y}}}} \right] = \mathop \sum \limits_{{\sum \alpha_{i} = 0}}^{\varvec{Q}} \left( {y_{\alpha } } \right)^{2}$. And so are the total Sobol indexes for each parameter $X_{i}$:

$$ I_{i}^{\text{PC}} = \frac{{\sum\nolimits_{{{\mathbf{u}} \in {\mathbf{u}}_{{\mathbf{i}}} }} {\left( {y_{{\mathbf{u}}} } \right)^{2} } }}{{\mathop \sum \nolimits_{{\sum \alpha_{i} = 1}}^{{\mathbf{Q}}} \left( {y_{\alpha } } \right)^{2} }} $$

(22)

where ${\mathbf{u}}_{{\mathbf{k}}} = \left\{ {\alpha :\alpha_{k} > 0} \right\}$ the set of $\alpha$ multi-indexes such that the indice $\alpha_{k}$ is non-null.

Another alternative to computing Sobol’ indexes is the the Direct Monte Carlo Method [12, 48]. This is particularly relevant when explicit model estimates are available but mostly when the PCE applicability is altered (case of discrete responses such as the cracking patterns evolution with the applied load). Otherwise, the use of PCE-based Global Sensitivity Analysis is recommended as it is considerably less time consuming (requiring less model calls compared to the Monte Carlo Methods).

Based on the ordering of $I_{i}^{\text{PC}}$ terms, one might be led to identify some negligible indexes $I_{i0}^{\text{PC}}$ compared to others. This informs about the negligible effect of the associated parameter $X_{i0}$ variation over the domain $D_{{X_{i0} }}$ on the computed response variance. The NDP can therefore be reduced—a posteriori—to $\left( {q + 1} \right)^{{N^{\prime} \le N}}$ where $N - N^{\prime}$ is the number of identified non-influential parameters.