A Proper Generalized Decomposition (PGD) approach to crack propagation in brittle materials: with application to random field material properties
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Abstract
Understanding the failure of brittle heterogeneous materials is essential in many applications. Heterogeneities in material properties are frequently modeled through random fields, which typically induces the need to solve finite element problems for a large number of realizations. In this context, we make use of reduced order modeling to solve these problems at an affordable computational cost. This paper proposes a reduced order modeling framework to predict crack propagation in brittle materials with random heterogeneities. The framework is based on a combination of the Proper Generalized Decomposition (PGD) method with Griffith’s global energy criterion. The PGD framework provides an explicit parametric solution for the physical response of the system. We illustrate that a nonintrusive samplingbased technique can be applied as a postprocessing operation on the explicit solution provided by PGD. We first validate the framework using a global energy approach on a deterministic twodimensional linear elastic fracture mechanics benchmark. Subsequently, we apply the reduced order modeling approach to a stochastic fracture propagation problem.
Keywords
Brittle fracture Crack propagation Model order reduction Proper Generalized Decomposition Random fields Monte Carlo method1 Introduction
One of the important goals in engineering design is to avoid catastrophic failure. Besides, in many applications, it is often crucial to understand the failure processes. To realistically model failure processes in engineering systems it is often essential to study the impact of uncertainties in the system parameters, such as loading conditions, specimen geometry, material properties, etc. Taking into account such uncertainties in an analysis typically implies that the number of times that a solution must be computed increases rapidly with an increase in the number of uncertain parameters. The use of reduced order models is then indispensable as these make it practical to solve the problem for many parameter realizations at an affordable computational effort.
While Reduced Order Modeling (ROM) is a wellestablished concept in the field of linear elastic solid mechanics [4, 6, 19], its application to fracture mechanics problems has remained essentially unexplored, with Ref. [25] providing a notable exception. In the present work, a new ROM technique for fracture propagation is presented which allows failure to be studied as a postprocessing operation of a parameterized solution that incorporates varying loads, crack lengths and material uncertainties. We propose a parameterization of the crack on the one hand, and a method to take into account the fracture propagation criterion in the reduced order model setting on the other hand. Furthermore, we extend the framework to include random heterogeneities in the material properties.
The reduction method of choice in this work is the Proper Generalized Decomposition (PGD) method, which is a reduced order modeling technique specifically designed to counter the curse of dimensionality induced by the increase in system parameters to be considered in an analysis [10]. The key idea of the PGD technique is to represent the generalized solution in the whole computational vademecum [28, 31] (i.e., the highdimensional parameter space) as a finite sum of terms that involve the product of functions of the system parameters. The computation of this generalized solution is referred to as the offline stage. Once the generalized solution has been obtained, the solution space can be browsed in a computationally efficient way, making it suitable for real time computations [8, 22]. This evaluation of the solution space for a particular set of system parameters is referred to as the online stage.
This paper is organized as follows. The model problem considered in this work is introduced in Sect. 2. Section 3 demonstrates how a separable form of the problem can be obtained in regard to the fracture length, which is a prerequisite for the application of the PGD method discussed in Sect. 4. We herein adapt the PGD formulation to solve a linear system of equations, which we refer to as the PGD solver [27]. Sect. 5 studies the accuracy of the fracture length parametrization in the setting of a stationary fracture. Section 6 then describes the application of the PGD framework to Griffith’s fracture model, along with the consideration of an LEFM benchmark test case [26]. Section 7 then presents an application in the stochastic setting, where we use the KarhunenLoève expansion [15, 23] to discretize random field material properties. A Monte Carlo based stochastic analysis is then performed that demonstrates the efficiency of the PGD framework. Conclusions are presented in Sect. 8.
2 Model fracture problem
A nonstandard aspect in relation to the fracture problem considered in this work, is that the crack length parameter, \(l_{c}\), enters the problem through the definition of the domain. As a consequence, the separable forms (8), with \(l_{c}\) as one of the parameters, will not follow naturally from (5). Obtaining separable forms instead requires recasting of the formulation in a canonical form through a pull back of the problem to a reference configuration. This reformulation of the problem is considered in the next section.
3 Fracture length parametrization
The linear system of equations corresponding to (14) is discretized using a finite element mesh constructed over the reference domain \({\Omega }^{\text {ref}}\). A regular, uniformly spaced, mesh is used, with an even number of elements in each direction (see Fig. 3). As a result, the boundary at \(X=0.5\), across which the mapping function (10) is nonsmooth, coincides with an element boundary. This has been found to be advantageous from an implementation point of view, as an element is either completely in the left side of the reference domain, \({\Omega }^{\mathrm{ref}}_\mathrm{left}=\{ \varvec{X}\in {\Omega }^{\text {ref}}\mid X \le 0.5 \}\), or completely in the right side of the reference domain, \({\Omega }^{\mathrm{ref}}_\mathrm{right}=\{ \varvec{X}\in {\Omega }^{\text {ref}}\mid X > 0.5 \}\). Although this particular choice of the referencedomain mesh is favorable from the vantage point of implementation and accuracy, the methodology presented herein is not restricted to this choice of the mesh, and could equally well be applied to unstructured meshes.
A fundamental difference between the finite element discretization over the reference grid, Eq. (14), and the system obtained using a direct discretization over the physical domain, equation (6), is that the crack length parameter in (14) appears inside the integrands of the matrix components, and not in the domain boundary (and constraints) definitions. This makes it possible to obtain the separable forms of the stiffness matrix and force vector required for the PGD framework.
4 The Proper Generalized Decomposition (PGD) method
The parametric problem (7) is solved here using the Proper Generalized Decomposition (PGD) method [2, 3, 8]. The particular use of the PGD method considered here follows the idea presented in [13, 27], where the method is applied to a discretized (in both space and parametric dimensions) system of linear equations. This differs from the standard use of PGD, where the method is applied to the weak form of the problem (e.g., [12, 24, 28, 31]).

The PGD algorithm requires the determination of separable forms of the stiffness matrix and force vector as input. As discussed in detail in Sect. 3, the discrete operator \(\mathbf {K}(l_{c})\) for the parametric problem with the crack length \(l_{c}\) as a parameter admits an exact separable representation. This is not generally the case, as we will discuss, for example, in the stochastic test case considered in Sect. 7. In situations where the linear system cannot be separated analytically, it is often replaced by a separable approximation (e.g., [30, 31]). There exist several methods to compute such separated approximations. For higherdimensional parameter domains various methods have been proposed in the literature, such as: an approximation based on the PGD concept [14], Singular Value Decomposition (SVD) type approximations [11], approximations based on the CANDECOMP/PARAFAC methods [7, 18], and Tucker decomposition type approximations [29]. An overview of these techniques can be found in, e.g., Ref. [21]. It is noted that in the case of highdimensional parameter domains, the computation of separable forms can be computationally demanding.
 A greedy algorithm [1, 8] is used to sequentially compute the terms to the PGD approximation \(\hat{\mathbf {u}}_{\text {pgd}}\) in Eq. (18). Given the PGD approximation with \(n_{pgd}1\) terms, here denoted byan enrichment term \(\hat{\mathbf {u}}^{n_{pgd}}\prod _{j=1}^{n_{\mu }}{G^{n_{pgd}}_j}\) is computed as to obtain the PGD approximation with \(n_{pgd}\) terms:$$\begin{aligned} \hat{\mathbf {u}}_{\text {pgd}}^{n_{pgd}1}(\varvec{\mu }) = \sum _{i=1}^{n_{pgd}1} \hat{\mathbf {u}}^i \prod _{j=1}^{n_{\mu }} {G^{i}_j}(\mu _j). \end{aligned}$$(21)Each enrichment term is computed one at a time, constructing the summation progressively until the convergence criterion$$\begin{aligned} \hat{\mathbf {u}}_{\text {pgd}}^{n_{pgd}} (\varvec{\mu })= \hat{\mathbf {u}}_{\text {pgd}}^{n_{pgd}1}(\varvec{\mu }) + \hat{\mathbf {u}}^{n_{pgd}}\prod _{j=1}^{n_{\mu }}{G^{n_{pgd}}_j}(\mu _j). \end{aligned}$$(22)is met with a userdefined tolerance of \(\epsilon _{glob}\). Each step in the greedy algorithm, i.e., computing each of the enrichment terms, involves the computation of the enrichment modes in space, \({\hat{\varvec{u}}}^i\) in discrete form, and in the parameter spaces, \(G_j^i(\mu _j)\). We herein compute these enrichments iteratively using an alternate direction solver, which is discussed in detail below.$$\begin{aligned} \frac{\beta ^{n_{pgd}}}{\beta ^{1}} = \frac{\Vert \hat{\mathbf {u}}^{n_{pgd}} \Vert \prod _{j=1}^{n_{\mu }} \Vert \hat{{\varvec{g}}}^{n_{pgd}}_j \Vert }{\Vert \hat{\mathbf {u}}^1 \Vert \prod _{j=1}^{n_{\mu }} \Vert \hat{{\varvec{g}}}^1_j \Vert } \le \epsilon _{glob}, \end{aligned}$$(23)

An alternating direction solution strategy [9] is used to compute the enrichment terms \(\hat{\mathbf {u}}^{n_{pgd}}\prod _{j=1}^{n_{\mu }}{G^{n_{pgd}}_j}\). Leveraging the separable forms, in this alternating direction strategy the spatial and parametric directions are treated sequentially as to reduce the higherdimensional parametric problem to a series of low dimensional problems. This iterative process is repeated until a fixed point is reached within a defined tolerance. For the explanation of this alternating direction strategy we will consider \(n_{\mu }=1\) with the fracture length \(\mu _1= l_{c}\) as the only parameter.
For the alternate direction solution strategy, the parametric problem (7) is considered in its weighted residual form:The unknowns in this system are the spatial and parametric enrichment modes, \(\hat{\mathbf {u}}^{n_{pgd}}\) and \(G^{n_{pgd}}_{l_{c}}(l_{c})\), respectively. The corresponding test functions are defined as:$$\begin{aligned}&\int _{{\mathcal {I}}_{l_{c}}} \delta \hat{{\varvec{v}}}( l_{c})^{\textsf {T}}\left[ \mathbf {K} (l_{c}) \left( \hat{\mathbf {{u}}}_{\text {pgd}}^{n_{pgd}1} (l_{c}) + \hat{\mathbf {u}}^{n_{pgd}} G^{n_{pgd}}_{l_{c}}( l_{c}) \right) \right. \nonumber \\&\quad \left.  \mathbf {f} (l_{c})\right] \,\mathrm{d}l_{c}= 0 \quad \forall \delta \hat{{\varvec{v}}}( l_{c}). \end{aligned}$$(24)In the alternate direction strategy, the system (24) is solved per spatial or parametric dimension:$$\begin{aligned} \delta \hat{{\varvec{v}}}( l_{c})= & {} \delta \left( \hat{\mathbf {u}}^{n_{pgd}} G^{n_{pgd}}_{l_{c}}( l_{c}) \right) = \delta \hat{\mathbf {u}}^{n_{pgd}} G^{n_{pgd}}_{l_{c}}( l_{c}) \nonumber \\&+ \hat{\mathbf {u}}^{n_{pgd}} \delta G^{n_{pgd}}_{l_{c}}( l_{c}). \end{aligned}$$(25)The above alternate direction steps are repeated until the relative difference between two successive steps is smaller than a prescribed tolerance, \(\epsilon _{local}\), Given an approximation (or initial guess) for the parametric enrichment mode \(G^{n_{pgd}}_{l_{c}}\), the system (24) reduces to the linear system:Using the separable forms for the stiffness matrix and force vector in equation (9), this system can be rewritten as$$\begin{aligned}&\int _{{\mathcal {I}}_{l_{c}}} G^{n_{pgd}}_{l_{c}}( l_{c}) \left[ \mathbf {K} (l_{c}) \left( \hat{\mathbf {{u}}}_{\text {pgd}}^{n_{pgd}1} (l_{c}) + \hat{\mathbf {u}}^{n_{pgd}} G^{n_{pgd}}_{l_{c}}( l_{c}) \right) \right. \nonumber \\&\quad \left.  \mathbf {f} (l_{c})\right] \mathrm{d}l_{c}= {\varvec{0}}. \end{aligned}$$(26)with \(n_k = 4\) and \(n_f = 2\). An essential idea of the PGD method is that the parametric integrals in this equation can be evaluated efficiently on account of the fact that these are lowdimensional integrals (in this particular case onedimensional). We herein use a standard trapezoidal integration rule for the evaluation of these integrals.$$\begin{aligned} \begin{aligned}&\left[ \sum _{i=1}^{n_k} \mathbf {K}^{i} \int _{{\mathcal {I}}_{l_{c}}} G^{n_{pgd}}_{l_{c}}( l_{c}) \phi ^{i}(l_{c}) G^{n_{pgd}}_{l_{c}}( l_{c}) \mathrm{d}l_{c}\right] \hat{\mathbf {u}}^{n_{pgd}} \\&\quad = \sum _{i=1}^{n_f} \mathbf {f}^{i} \int _{{\mathcal {I}}_{l_{c}}} G^{n_{pgd}}_{l_{c}}( l_{c}) \psi ^{i}(l_{c})\mathrm{d}l_{c}\\&\qquad  \sum _{i=1}^{n_k} \mathbf {K}^{i} \int _{{\mathcal {I}}_{l_{c}}} G^{n_{pgd}}_{l_{c}}( l_{c}) \phi ^{i}(l_{c}) \hat{\mathbf {{u}}}_{\text {pgd}}^{n_{pgd}1} (l_{c}) \mathrm{d}l_{c}. \end{aligned} \end{aligned}$$(27)
 Given the spatial enrichment mode \(\hat{\mathbf {u}}^{n_{pgd}}\) computed through the system (27), the parametric enrichment mode \(G^{n_{pgd}}_{l_{c}}\) can be obtained from the system (24). From (24) it follows that for all \(\delta G^{n_{pgd}}_{l_{c}}( l_{c})\):Equivalently, it holds that for each fracture length \(l_{c}\)$$\begin{aligned} \begin{aligned}&\int _{{\mathcal {I}}_{l_{c}}} \delta G^{n_{pgd}}_{l_{c}}( l_{c})\left[ \left( {\hat{\varvec{u}}}^{n_{pgd}} \right) ^{\textsf {T}} \mathbf {K} (l_{c}) \left( \hat{\mathbf {{u}}}_{\text {pgd}}^{n_{pgd}1} (l_{c}) \right. \right. \\&\quad \left. \left. + \hat{\mathbf {u}}^{n_{pgd}} G^{n_{pgd}}_{l_{c}}( l_{c}) \right)  \mathbf {f}(l_{c})\right] \,\mathrm{d}l_{c}= 0. \end{aligned} \end{aligned}$$(28)from which the parametric enrichment mode follows directly as:$$\begin{aligned} \begin{aligned}&\left[ \left( {\hat{\varvec{u}}}^{n_{pgd}} \right) ^{\textsf {T}} \mathbf {K} (l_{c}) \left( \hat{\mathbf {{u}}}_{\text {pgd}}^{n_{pgd}1} (l_{c})\right. \right. \\&\quad \left. \left. + \hat{\mathbf {u}}^{n_{pgd}} G^{n_{pgd}}_{l_{c}}( l_{c}) \right)  \mathbf {f} (l_{c})\right] = 0, \end{aligned} \end{aligned}$$(29)Substitution of the separable forms for the stiffness matrix and force vector then finally yields:$$\begin{aligned} G^{n_{pgd}}_{l_{c}}( l_{c}) = \frac{\left( {\hat{\varvec{u}}}^{n_{pgd}} \right) ^{\textsf {T}} \left( \mathbf {f} (l_{c})  \mathbf {K} (l_{c}) \hat{\mathbf {{u}}}_{\text {pgd}}^{n_{pgd}1} \right) }{ \left\ {\hat{\varvec{u}}}^{n_{pgd}} \right\ ^2 }. \end{aligned}$$(30)This expression for the parametric enrichment mode can be evaluated quickly by virtue of the fact that the dimensions are separated in the sense that it is not required to reassemble the finite element system for each fracture length. The parametric enrichment mode is represented discretely by projection onto the parametric basis in Eq. (19). Since this discretization pertains to a linear finite element basis, the coefficients \(\hat{{\varvec{g}}}^{n_{pgd}}_{l_{c}}\) can be computed by evaluation of Eq. (31) in the parametric nodes.$$\begin{aligned}&G^{n_{pgd}}_{l_{c}}( l_{c})\nonumber \\&\quad = \frac{\left( {\hat{\varvec{u}}}^{n_{pgd}} \right) ^{\textsf {T}} \left( \sum _{i=1}^{n_f} \mathbf {f}^{i}\psi ^{j}(l_{c})  \sum _{i=1}^{n_k} \phi ^{i}(l_{c}) \mathbf {K}^{i} \hat{\mathbf {{u}}}_{\text {pgd}}^{n_{pgd}1} \right) }{ \left\ {\hat{\varvec{u}}}^{n_{pgd}} \right\ ^2 }. \nonumber \\ \end{aligned}$$(31)
with the subscript iter denoting the alternate direction step, and with the norms defined as:$$\begin{aligned} \frac{\left\ \left. {\hat{\varvec{u}}}^{n_{pgd}} G_{l_{c}}^{n_{pgd}} (l_c) \right _{iter+1}  \left. {\hat{\varvec{u}}}^{n_{pgd}} G_{l_{c}}^{n_{pgd}} (l_c) \right _{iter} \right\ }{\left\ \left. {\hat{\varvec{u}}}^{n_{pgd}} G_{l_{c}}^{n_{pgd}} (l_c) \right _{iter+1} \right\ } < \epsilon _{local}, \end{aligned}$$(32)$$\begin{aligned} \left\ {\hat{\varvec{u}}}^{n_{pgd}} G_{l_{c}}^{n_{pgd}} (l_c) \right\ = \left\ {\hat{\varvec{u}}}^{n_{pgd}} \right\ \int _{{\mathcal {I}}_{l_{c}}}  G_{l_{c}}^{n_{pgd}} (l_c) \mathrm{d}l_{c}. \end{aligned}$$(33) 
5 Numerical analysis of the PGD approximation behavior
Convergence study parameter settings
Domain width  \(H_x\)  4  m 
Domain height  \(H_y\)  4  m 
Young’s modulus  E  1  GPa 
Poisson ratio  \(\nu \)  0.1  
Traction on top boundary  \(\varvec{t}\)  (0, 100)  MPa 
Parameter domain  \({\mathcal {I}}_{l_{c}}\)  [1,3]  m 
Enrichment tolerance  \(\epsilon _{glob}\)  \(10^{3}\)  
Fixedpoint tolerance  \( \epsilon _{local}\)  \(10^{6}\) 
In the setting considered here, the separable form derived in Sect. 3 is exact up to integration accuracy. Since the integrals are herein evaluated with Gauss schemes of sufficiently high degree, the separable forms are accurate up to floating point precision. In general, however, the separable form (9) is not exact, as we will consider, for example, in the context of the stochastic analysis presented in Sect. 7. An important first step in studying the approximation behavior of the PGD approximation is then to study the accuracy of the separable form (9). This accuracy can be assessed by comparison of the matrix and right hand side obtained through the separable form (9) with their corresponding original finite element counterparts. Evidently, one has to perform this accuracy assessment in such a way that the parameter variations admitted by the PGD expansion are properly taken into account.
5.1 Spatial mesh size dependence
5.2 Parametric mesh size dependence
All results presented above were based on a fixed parametric mesh size of \(h_{l_{c}}\approx 0.015\) and variations in the spatial mesh size. We now consider the influence of variations in the parametric mesh size under a fixed spatial mesh size of \(h=0.0625\) m.
Figure 8 shows that both the parameterdependent energy error (34) and mean energy error (35) are virtually independent of the parametric mesh size even on parametric meshes as coarse as \(h_{l_{c}}=0.125\) m (8 elements). This conveys that, in the setting considered here, the accuracy is governed by the number of PGD modes rather than by the resolution of the parametric mesh.
6 Application of the PGD framework to propagating fractures
6.1 The fracture propagation criterion
We consider Griffith’s model [16] for crack propagation in brittle materials. The conceptual idea of this model is that a fracture will propagate if the energy stored in the material is sufficiently large to overcome the fracture energy associated with the creation of new fracture surface. For linear elastic materials an equivalent interpretation of this energybased model is provided through the concept of stress intensity factors [5]. In the context of the PGD framework we find the energy perspective most suitable, as it provides the possibility to evaluate the propagation criterion directly based on the parametric solution (37).
6.2 Numerical example: a centercrack under tensile loading
6.2.1 Stress intensity factors
Figure 9 shows the dimensionless stress intensity factors \({\mathcal {K}}_1/{\mathcal {K}}_0\) for various parameter configurations, i.e., different \(l_{c}/H_x\) and \(H_x/H_y\) (see Ref. [26] for a benchmark result). Note that the plotted factors are nondimensionalized using \( {\mathcal {K}}_0 = (\lambda {\hat{\varvec{t}}} \cdot \varvec{n}) \sqrt{\pi l_{c}}\), where \(\lambda {\hat{\varvec{t}}} \cdot \varvec{n}\) gives the magnitude of the applied tensile traction. Figure 9 compares the PGD results based on the settings mentioned in Table 1 for a mesh size \(h=0.0625\) m. However, note that this plot of nondimensional stress intensity factors is independent of the input values, i.e., even for different values of geometry and load, similar curves for \({\mathcal {K}}_1/{\mathcal {K}}_0\) are obtained. This figure conveys that for the given PGD settings, the stress intensity factor can be computed accurately using the PGD expansion (37). While each point in Fig. 9 would typically represent a finite element simulation in the traditional FEM setting, in the PGD case these are all mere evaluations of the expansion.
6.2.2 Fracture propagation
 1.The region where the crack is stable:$$\begin{aligned}&\frac{\partial {{\mathcal {E}}}}{\partial l_{c}}< 0&\text{ or }&{\mathcal {G}}(l_{c},\lambda ) < {\mathcal {G}}_c. \end{aligned}$$
 2.The region where the energy balance is critical:$$\begin{aligned}&\frac{\partial {{\mathcal {E}}}}{\partial l_{c}} = 0&\text{ or }&{\mathcal {G}}(l_{c},\lambda ) = {\mathcal {G}}_c. \end{aligned}$$
 3.The unstable propagation region:$$\begin{aligned}&\frac{\partial {{\mathcal {E}}}}{\partial l_{c}}> 0&\text{ or }&{\mathcal {G}}(l_{c},\lambda ) > {\mathcal {G}}_c. \end{aligned}$$
For a particular load scale, until the critical point is reached the crack is stable (green region in Fig. 10a), and beyond the maximum point the crack is unstable (red region in Fig. 10a). The critical energy states are connected in the form of a curve which gives the critical load value for each fracture length. This curve can be identified in Fig. 10a as the line separating the green area from the red area. The key insight is to recognize that, for a shorter crack length, which is left of the critical value point, the total energy (43) of the system increases with increasing crack length. Therefore, additional energy must be stored into the material before the crack can propagate, and hence the crack is stable. However, at longer crack lengths, which is right of the maximum value, an increase in crack length leads to a decrease in total energy, which therefore leads to unstable crack propagation. Evidently, the loadbearing capacity of the specimen decreases as the fracture propagates.
7 Application to fracture propagation in random heterogeneous materials
On account of (45) the Young’s modulus at any fixed location, \({\tilde{E}}(\varvec{x})\), is normally distributed. The variation \(\sigma _{E}^2\) is selected such that physically impossible negative realizations are avoided. Although not considered herein, the PGD framework can be applied without modification to, e.g., lognormal random fields. It is noted that we herein construct the random field over the computational domain, thereby implicitly assuming that the random material properties adhere to the symmetries of the homogeneous problem. Preservation of the symmetries is in line with the considered parametrization of the fracture problem, as nonsymmetries would result in deviations of the fracture path from the xaxis. Although such variations are evidently physical, consideration of these within the PGD framework is beyond the scope of this manuscript.
7.1 Separable representation of the random system of equations
7.2 Monte Carlo analysis of the critical load
Using the separable form for the stiffness matrix as discussed in Sect. 7.1, the PGD solver discussed in Sect. 4 is used to attain the PGD solution (47). We here use this parametrized solution to perform a Monte Carlo simulation to attain the probability distribution and statistical moments of the critical loading force for specimens with various initial fracture lengths.
To construct the PGD solution (47) it is necessary to consider a finite dimensional domain for the random parameters, \(\tilde{{\varvec{z}}}\), which parametrize the Karhunen–Loève expansion for the Young’s modulus (45). We herein truncate the random domain to \({\mathcal {I}}_{{\tilde{z}}_i}=[5,5]\) for \(i=1,\ldots ,n_{kl}\), based on the idea that realizations beyond this range are unlikely and will have a minor effect on the mean and standard deviation of the critical force. We generate realizations of the uncorrelated random variables \(\tilde{{\varvec{z}}}\) using a random number generator, and we discard realizations outside of the truncated random domain.
In a typical FEbased Monte Carlo simulation, evaluation of the critical loads is computationally demanding, which practically restricts the sample sizes that can be considered. Therefore, in such cases, a sample size is selected that strikes an adequate balance between the confidence level of the attained statistical moments and the required computational effort. In the PGD setting considered here, the computational effort involved in determining the critical force for a given realization of the random field is negligible compared to the corresponding full finite element simulation. This allows for the consideration of sample sizes that are orders of magnitude larger than those that could be considered using direct FE analysis, which in turn enables the computation of the statistical moments with confidence levels that are practically beyond the reach of direct FE analyses. Evidently, the selection of the sample size should be based on a tradeoff between the error in the PGD approximation and the confidence level of the Monte Carlo method.
7.3 Numerical example: a centercrack under tensile loading
We consider a Karhunen–Loève discretization consisting of \(n_{kl}=3\) modes, which are shown in Fig. 11. In Fig. 12 we show two realizations of the KL expansion, as well as a samplingbased reconstruction of the autocorrelation function (46). On account of the low spatial frequency of the variations, the KL expansion with only 3 terms is observed to already appropriately reproduce the autocorrelation function.
Figure 14 displays the probability distribution of the critical load for various settings of the initial crack length. The displayed results are based on a sample size of 5000. Note that for each of the displayed subplots in Fig. 14 a separate Monte Carlo simulation is required, which would be computationally impractical using a direct FE approach. The efficiency with which realizations can be computed from the PGD approximation (47) allows us to perform Monte Carlo analyses for different settings in the parameter space. This results, for example, in the evaluation of the critical force versus the initial crack length as displayed in Fig. 15a. The confidence level of the mean values displayed in this plot is approximately 98% based on a sample size of 5000 realizations. Such confidence levels are impractical to obtain using direct FE Monte Carlo.
Figures 14 and 15 show that the average critical load bearing capacity decreases with an increase in crack length, while a decrease in the standard deviation is observed. The deterministic result is plotted for reference, from which it is observed that the computed mean is slightly smaller than the deterministic value. The observed results from the Monte Carlo simulation are in good agreement with perturbation analysis results (see [17] for an overview) based on the analytical fracture loads for homogeneous specimens, which is to be expected on account of the considered low spatial frequency of the random input.
 a.
The realization closest to the mean fracture load of 77.5 MN corresponds to a Young’s modulus field which is very close to its mean value everywhere in the specimen.
 b.
The realization with the largest fracture load of 88.5 MN corresponds to a Young’s modulus field which is very high throughout the specimen (on average approximately 25% higher than its mean value), and is particularly large near the tip of the initial crack.
 c.
The realization with the smallest fracture load of 66.6 MN corresponds to a Young’s modulus field which is very low throughout the specimen (on average approximately 25% lower than its mean value), and particularly near the tip.
8 Conclusions
In this work we have proposed a reducedorder modeling technique for a prototypical linear elastic fracture mechanics problem. An essential ingredient in the proposed approach is to introduce the parametrization of the crack through a geometric mapping. For the considered model problem it then follows that a separable form of the stiffness matrix and external force vector can be obtained analytically, which makes it possible to apply the Proper Generalized Decomposition method to obtain a solution to the parametric problem.
The suitability and performance of the proposed framework is demonstrated using a series of numerical test cases, starting with a convergence study for the parametric decomposition. This study conveys that the introduced geometric mapping function for the fracture parameter behaves in accordance with the wellunderstood behavior of the PGD framework. The PGD fracture framework is further demonstrated using two propagating fracture test cases.
In the first test case it is demonstrated how Griffith’s propagation criterion can be evaluated efficiently using the PGD approximation. The representation of the fracture length in the PGD solution enables the straightforward computation of the energy release rate, which is in contrast with standard finite element methods, which generally require dedicated numerical techniques for the evaluation of the corresponding shape derivative.
In the second test case the PGD approximation is used to efficiently perform a fracture analysis in the presence of random material heterogeneities. Using a singular value decomposition for the interpolation of the random field of elastic properties pulled back to the reference configuration, an approximate separable form of the stiffness matrix is obtained. The random variable coefficients of the Karhunun–Loève field for the modulus of elasticity appear as parameters in this separable form. Since the fracture load can be computed as a postprocessing operation on the PGD approximation, MonteCarlo simulations can be performed with sample sizes (and confidence levels) that are beyond the typical reach of direct samplingbased stochastic finite element analyses.

While the considered fracture is parametrized by a single variable, namely the fracture length, this is evidently not possible in the case of more complex fractures. Of course, the range of applicability of the proposed technique can be extended to a reasonably sized class of fracture problems using a relatively low dimensional parameter space for the fracture geometry. Think for example of slanted fractures in plane strain or plane stress settings, which, besides the length, would require the fracture angle as an additional parameter. In general, however, representing more complex fracture geometries will rapidly increase the number of parameters, which is detrimental to the performance of the PGD framework. This is particularly the case when one opts to consider a piecewise representation of fractures, which is natural to finite element methods.

For more complex fracture patterns, constructing a suitable geometric mapping function will be considerably more challenging than in the prototypical benchmark considered in this work. Constructing a mapping analytically is very restrictive, but it is very well imaginable that one can construct discrete mapping operators (mapping nodal reference coordinates to nodal physical coordinates). Such more advanced mappings – the construction of which evidently warrants further investigation – will, however, pose several difficulties. For example, the analytical separation of the system of equations as obtained in this work will not be generally obtainable, which hence requires the consideration of potentially computationally demanding approximations for the separable forms. Moreover, an open research question remains how to deal with fractures with changing topology (e.g., branching, merging), as topological changes can in general not be captured by the proposed mapping technique.
Notes
Acknowledgements
We acknowledge the support from the European Commission EACEA Agency, Framework Partnership Agreement Erasmus Mundus Action 1b, as a part of the EM Joint Doctorate Simulation in Engineering and Entrepreneurship Development (SEED). The work of S. Zlotnik and P. Díez was funded by the project DPI201785139C22R of the Spanish Ministry and by grant 2017SGR1278 from the Generalitat de Catalunya.
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