1 Introduction
2 Model fracture problem
3 Fracture length parametrization
4 The Proper Generalized Decomposition (PGD) method
- 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 higher-dimensional 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 high-dimensional 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 user-defined 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 higher-dimensional 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 low-dimensional integrals (in this particular case one-dimensional). 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
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 | \(\epsilon _{glob}\) | \(10^{-3}\) | |
Fixed-point tolerance | \( \epsilon _{local}\) | \(10^{-6}\) |
5.1 Spatial mesh size dependence
5.2 Parametric mesh size dependence
6 Application of the PGD framework to propagating fractures
6.1 The fracture propagation criterion
6.2 Numerical example: a center-crack under tensile loading
6.2.1 Stress intensity factors
6.2.2 Fracture propagation
7 Application to fracture propagation in random heterogeneous materials
7.1 Separable representation of the random system of equations
7.2 Monte Carlo analysis of the critical load
7.3 Numerical example: a center-crack under tensile loading
8 Conclusions
- 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.