Coupled domain decomposition–proper orthogonal decomposition methods for the simulation of quasi-brittle fracture processes
- Alberto Corigliano^{1}Email authorView ORCID ID profile,
- Federica Confalonieri^{1},
- Martino Dossi^{1} and
- Stefano Mariani^{1}
DOI: 10.1186/s40323-016-0081-9
© The Author(s) 2016
Received: 6 February 2016
Accepted: 21 October 2016
Published: 14 November 2016
Abstract
In this paper, we discuss a strategy to reduce the computational costs of the simulation of dynamic fracture processes in quasi-brittle materials, based on a combination of domain decomposition (DD) and model order reduction (MOR) techniques. Fracture processes are simulated by means of three-dimensional finite element models in which use is made of cohesive elements, introduced on-the-fly wherever a cracking criterion is attained. The body is initially subdivided into sub-domains; for each sub-domain MOR is obtained through a proper orthogonal decomposition (POD) of the equations governing its evolution, until when it starts getting cracked. After crack inception within a sub-domain, the solution is switched back to the original full-order model for that sub-domain only. The computational gain attained through the coupled use of DD and POD thus depends on the geometry of the body, on the topology of sub-domains and, on top of all, on the spreading of cracking induced by load conditions. Numerical examples concerning well-established fracture tests are used for validation, and the attainable reduction of the computing time is discussed at varying decomposition into sub-domains, even in the absence of a full exploitation of parallel computing potentialities.
Keywords
Quasi-brittle fracture Domain decomposition Model order reduction (MOR) Proper orthogonal decomposition (POD)Background
One of the most active sectors in computational mechanics looks for more and more efficient strategies for the highly realistic simulation of complex phenomena. Two main driving forces guide these researches. The first one is the attempt to simulate phenomena in an extremely fast way, possibly reaching real-time computing; recent contributions in this field are e.g. [1, 2]. The second strong motivation is the attempt to perform simulations that with standard strategies would simply be practically impossible due to unacceptable computing times; examples in this line are those of [3–5].
Domain decomposition (DD) and model order reduction (MOR) approaches are typical tools developed to reach the above goals, not only in the field of computational mechanics but more generally in the extremely vast field of computational sciences. Proper orthogonal decomposition (POD), see e.g. [6], is a MOR strategy that has recently received much attention as a tool to drastically reduce the number of degrees of freedom (DOFs) retained in an analysis, being extremely efficient in linear problems. Many difficulties arise when trying to apply POD to non-linear, irreversible problems; various strategies have been recently developed to this purpose, among these e.g. the so-called proper generalized decomposition (PGD) originally proposed by Ladevèze with a different terminology and recently applied e.g. in [7–10].
The purpose of this paper is to further contribute to a series of works recently published by the Authors on the efficient application of DD and MOR-POD techniques to multi-physics and non-linear, irreversible problems. DD methods, based on an extension of the algorithm originally proposed in [11], were applied to the solution of the electro-mechanical coupled problem in microsystems in [12], and to the simulation of quasi brittle fracture processes in [13]. A combination of DD and MOR-POD techniques was proposed in [14] for the electro-mechanical coupled problem in microsystems and in [15] for the solution of elasto-plastic dynamic problems.
Moving from the aforementioned implementation in [15], we address here the problem of dynamic propagation of cohesive cracks in a quasi-brittle material. Although the spreading of dissipative phenomena can be rather different in elasto-plasticity and quasi-brittle cracking, especially when the solution is dominated in the latter case by the growth of a dominant crack, the two problems can be approached in a similar way as far as MOR is concerned. The coupled DD-POD approach here adopted is therefore similar to the one proposed in [15]: the whole body is initially subdivided in sub-domains; for each sub-domain, a reduced-order model is trained in the initial stage of the analysis; as soon as a sub-domain starts being cracked, the relevant solution is switched back to the original full-order model and then advanced in time. The handling of each sub-domain through an either reduced- or full-order mode can be therefore optimally approached via an heterogeneous time integration procedure, see [13].
The proposed approach reaches a good compromise between the contrasting needs of realistically reproducing complex fracture processes in possibly highly heterogeneous materials, and of being able to keep under acceptable levels the computing time.
The paper is organized in three main sections in addition to the “Introduction” and “Conclusion” sections. First, the semi-discretized problem formulation for an elastic body containing cohesive fracture process zones, in the presence of dynamic loading is discussed and presented. In the central main section the proposed new computational strategy is described in details. The third main section is devoted to the critical discussion of num erical examples.
Throughout the paper, a matrix Voigt notation is adopted.
Problem formulation
The governing equations are completed by the initial conditions in terms of displacement and velocity fields.
Reduced-order modelling approach
With traditional 3D finite element simulations of cracking in micro-structured solids under impact loading, the relevant computational burden can be extremely high. Using e.g. an explicit time stepping scheme ruled by the Courant–Friedrichs–Levy condition, the time step size typically becomes very small since a very refined spatial discretization is required to account for microstructural features (like the grain morphology) and to be compliant with condition (10). A reduced-order modelling approach is introduced next, coupling DD and POD strategies.
According to the DD algorithm proposed in [13, 29], the procedure presented here exploits the decomposition into smaller problems relevant to the sub-domains, avoiding the time-step limits imposed by a standard explicit scheme for the whole body. Similarly to the algorithm proposed in [15] for the elastic-plastic structural problem, we initially adopt an implicit integration scheme for all the sub-domains; whenever a sub-domain gets traversed by a crack, the integration of the relevant equations of motion of that sub-domain switches to an explicit one and the algorithm becomes multi-time-step.
The mechanical response of each linear elastic sub-domain is integrated in time with the Newmark average acceleration scheme, while the central difference scheme is used in the cracked ones, see [39]. After an initial synchronous phase, the presence of two time scales is handled when fracture starts to propagate: a coarse time step size \(\Delta t_{imp}\) is assigned to the sub-domains not crossed by cracks, and a fine step size \(\Delta t_{exp}= \frac{1}{m} \Delta t_{imp}\), with m integer, is instead associated to those crossed by cracks, as originally proposed in [11, 13] and schematically shown in Fig. 3.
Let us introduce the time discretization in Eq. (14), focusing on the time interval \([t_n,t_{n+1}]\). Because of the two-time-scales algorithm here exploited (see Fig. 3), the free and link problems have to be solved only once at time \(t_{n+1}\) for the linear implicit sub-domains (integrated with the coarse time scale) and at each intermediate time instant \(t_j\) with \(t_{n} \le t_{j} \le t_{n+1}\) for the explicit non linear sub-domains (integrated with the fine time scale).
The generalized multi-time step explicit–implicit method in the case of multiple sub-domains \((s=1,2,\ldots , n_{sd})\) is described in Algorithm 1. \(\Delta t_{ref}\) stands for the time step of the reference time scale. According to the formulation proposed in [11], such reference time scale is characterized by the smallest time step of the analysis (i.e. the implicit time step in the initial synchronous stage and the explicit time step after crack initiation).
Matching meshes at the interfaces between adjacent sub-domains are considered. This assumption allows to guarantee that the numerical dissipation due to the DD approach, basically linked to the work of the interface forces, does not sensibly affect the energy balance of the system, see [11, 13].
The above described DD strategy allows to reduce the computational burden but, in order to remarkably speedup the simulations, a POD-based reduced order modelling scheme is also allowed for. POD, in its snapshot version [43], is initially adopted to reduce the order of the whole problem. Next, as soon as a crack gets incepted in one sub-domain, its numerical model is switched back to the full-order one to properly account for energy dissipation in the process zone and for the changing topology of the crack surface \(\Gamma _c\); for all the sub-domains within which cracks are not triggered, the solution of the reduced-order equation of motion is still advanced in time with an implicit scheme.
It is well-known that the snapshot version of POD needs a training stage at the beginning of the analysis, to set the reduced-order model. During this training phase, the bases of the reduced-order space, onto which the equations of motion must be projected, are defined. In this work the duration of such training stage is heuristically determined (see [44]) and it is the same for all the sub-domains; otherwise adaptive procedures controlling the convergence of the bases towards a steady-state solution can be adopted. The latter approach was shown in [14, 15] to provide a further speedup to coupled DD-POD analyses, since convergence can be attained at different time instants in different sub-domains, and so it is not the late sub-model to govern by itself the duration of the training stage.
Since we assume that the whole body behaves elastically up to fracture initiation, the algorithm proposed in [13, 29] is used during the training phase to compute the snapshots for each sub-domain. For the problem under study, snapshots do not consist only of the time evolving response of the body to the external loads, but also of the inter-domain continuity. As said, in this phase an implicit, synchronous time stepping procedure is adopted. In [15, 42], a thorough discussion of the rationale followed for reduced-order modelling of nonlinear problems was presented; in what follows, only the details relevant to the coupled use of POD and DD, and to the algorithmic handling of the switch from the reduced-order model to the full-order one for the cracked sub-domains are summarized.
The singular values and relevant vectors are usually associated to the (oriented) energy content of the system under the given excitation. For structural problems, one may assume that if matrix \(\mathbf {S}_s\) collects nodal displacements information is obtained concerning the (elastic) energy stored in the bulk of system, whereas if the same matrix collects nodal velocities information is obtained about the system kinetic energy. As the total energy of the system is conserved in the absence of dissipative phenomena (namely, if crack evolution does not take place), the two aforementioned energy terms are related. Anyhow, as pointed out in [46], associating the singular values to the actual elastic and kinetic energies of the system is ”incorrect in principle and may yield misleading results.” So, to ensure the stability and invertibility of the interface operator (16), see also [47], in the snapshot matrix nodal displacements are gathered, and a weak continuity across sub-domains is enforced through the local stiffness \(\mathbf {k}\), see Eq. (15).
Numerical examples
Two examples are discussed in this section with the aim to show the potentialities of the proposed approach. The first one is a double cantilever beam (DCB) in which a mode-I crack process is characterized by initiation, propagation and possible arrest along a plane. The second example concerns the mixed-mode crack propagation, with a crack-path which deviates during the fracture process.
The approach described has been implemented in Fortran 90, and the simulations have been run on a PC featuring an Intel CoreTM i7-2600 CPU @3.4 GHz with a 64 bit operating system. The provided computational gain has been computed as \(\frac{T_{r-o}-T_{f-o}}{T_{f-o}}\), where \(T_{r-o}\) and \(T_{f-o}\) are the run times (CPU times) corresponding to the full-order (DD only) and reduced-order analyses, respectively. \(T_{r-o}\) gathers both the CPU time required for the training of the ROMs, and the CPU time to advance in time the solution for all the sub-domains. While the training of the ROMs can be optimized through a so-called thin SVD procedure [15, 51] in place of the standard one in Eq. (21), the duration of the reduced-order analysis is obviously affected by \(t_{end}\) (see Algorithm 1). Results collected next thus have to be considered representative, to also show how the selected number of sub-domains can affect the gain.
As mentioned before, the solution for each elastic sub-domain is advanced in time with the Newmark average acceleration scheme featuring \(\gamma _s = 1/2\) and \(\beta _s = 1/4\), \(\gamma _s\) and \(\beta _s\) being coefficients of the Newmark’s time integration algorithm; while the equations related to the cracked ones are integrated in time with the central difference scheme featuring \(\gamma _s = 1/2\) and \(\beta _s = 0\). In the reduced order simulations, we have always adopted \(\eta _s = 0.999\) for each sub-domain to ensure high accuracy of the solutions.
Double cantilever beam
In this section, a dynamic DCB test [52, 53] is considered. The specimen geometry is shown in Fig. 5, and the material properties are those proposed in [25], see Table 1. The beam length L is taken equal to 12 mm, so that the crack propagation is not influenced by the wave reflections from the right end of the beam. The beam has a rectangular cross section of width B equal to 0.1 mm and height 2 h equal to 0.2 mm. The initial crack length a is taken equal to 0.4 mm.
Mechanical properties of alumina adopted in the simulations
Property | Symbol | Value |
---|---|---|
Young’s modulus (\(\mathrm {GPa}\)) | E | 260 |
Mass density (\(\mathrm {kg/m^3}\)) | \(\rho \) | 3600 |
Poisson’s ratio (–) | \(\nu \) | 0.21 |
Maximum tensile strength (\(\mathrm {MPa}\)) | \(\tau _{\text {max}}\) | 400 |
Fracture energy (\(\mathrm {N/m}\)) | G | 34 |
The experimental test is performed by slowly opening the beam by means of a wedge and then registering the dynamic crack propagation. The initial conditions are here reproduced, performing a static elastic analysis under an imposed crack tip opening displacement \(V_0\). The crack tip is then released and a dynamic analysis is run. Two different testing conditions, in the following referred to as A and B, have been simulated considering \(V_0=4\) \(\upmu \)m and \(V_0=14\) \(\upmu \)m respectively. According to the analytical solutions proposed, for instance, in [52, 53], in the first case the crack stops propagating after reaching a plateau value, whereas an unstable fracture propagation is expected in the second case.
Double cantilever beam test: number of degrees of freedom and elements corresponding to each sub-domain, for the adopted domain decompositions (see Fig. 6b)
Partition | Degrees of freedom | Elements | ||||||
---|---|---|---|---|---|---|---|---|
DD(2sd) | 237,924 | 23,559 | 53,713 | 4183 | ||||
DD(2sd)-POD | 237,924 | 23,559 | 53,713 | 4183 | ||||
DD(3sd)-POD | 112,338 | 127,059 | 23,559 | 25,065 | 28,648 | 4183 | ||
DD(4sd)-POD | 68,697 | 87,624 | 83,979 | 23,559 | 15,146 | 19,699 | 18,868 | 4183 |
Figure 7 shows the crack tip position history for case A. Two different stages can be identified: in the first one, roughly until 0.2 ms, the crack propagation is governed by the stress wave propagation, while in the second one it is determined by a beam-like behavior. Finally, the crack arrests at about 0.7 ms. The crack is always confined in the first sub-domain for all the partitions, except for the 4 sub-domains case, in which also the neighbour sub-domain is reached by the fracture propagation.
To check the performance of the proposed DD-POD algorithm, the numerical results obtained with it are compared with the curve obtained with a domain decomposition simulation performed with a 2 sub-domains partition. In addition, the comparison with the reference analytical solution developed in [53], where the Bernoulli–Euler beam theory is employed to model the upper and the lower arm as beams of evolving length, is shown. A noteworthy good agreement of the results can be observed for all the listed simulations even if some discrepancies can be detected between the analytical solution and the numerical ones in the first part of the curve, due to the fact that the approximated analytical solution is not able to describe the effect of the elastic waves on crack propagation. Independently of the adopted DD, see also [13], the final lengths at crack arrest differ from the analytical solution by no more than 13%.
Double cantilever beam test—case A: run time, overall error and computational gain with respect to the DD approach
Run time (s) | Error w.r.t. DD (–) | Gain w.r.t. DD (%) | |
---|---|---|---|
DD | 34, 970 | – | – |
DD(2sd)-POD | 32,642 | \(2.2 \cdot 10^{-3}\) | \(-6.7 \) |
DD(3sd)-POD | 23,950 | \(1.3 \cdot 10^{-3}\) | \(-31.5\) |
DD(4sd)-POD | 27,591 | \(1.4 \cdot 10^{-3}\) | \(-21.1\) |
Double cantilever beam test—case B: run time, overall error and computational gain with respect to the DD approach
Run time (s) | Error w.r.t. DD (–) | Gain w.r.t. DD (%) | |
---|---|---|---|
DD | 35, 442 | – | – |
DD(2sd)-POD | 33,386 | \(1.8 \cdot 10^{-3}\) | \(-5.8 \) |
DD(3sd)-POD | 27,822 | \(3.3 \cdot 10^{-3}\) | \(-21.5\) |
DD(4sd)-POD | 28,602 | \(4.5 \cdot 10^{-3}\) | \(-19.3\) |
Tables 3 and 4 report the results of the total run time for each simulation performed for case A and B respectively. The different sub-divisions of the domain lead to very different computational gains, evaluated with respect to the DD algorithm proposed in [13, 29]. As expected, the gain in the case of the 2 sub-domains partition with the DD-POD algorithm is almost negligible, because of the small size of the uncracked subdomain, which collects the coarse size elements at the right side of the body. It could be noticed that the maximum gain is achieved whenever the fracture crosses the minimum number of subdomains, ideally only one subdomain. In such case, a computational gain up to 31% with respect to the DD algorithm proposed in [13, 29] is obtained.
Figures 9 and 10 show the contour plots of the \(\sigma _x\) component (x being the direction the beam axis) of the stress vector on the deformed configuration obtained with the reference DD numerical procedure and with the DD-POD algorithm, for case A and B respectively; the good agreement featured by the outcomes of all the simulations, testifies the accuracy of the kinematic fields modelled by the ROMs. Notice that the displacements are amplified to better display the deformed configuration.
Edge-cracked plate under impulsive loading
Mechanical properties of 18Ni1900 steel adopted in the simulations
Property | Symbol | Value |
---|---|---|
Young’s modulus (\(\mathrm {GPa}\)) | E | 190 |
Mass density (\(\mathrm {kg/m^3}\)) | \(\rho \) | 8000 |
Poisson’s ratio (–) | \(\nu \) | 0.3 |
Maximum tensile strength (\(\mathrm {MPa}\)) | \(\tau _{\text {max}}\) | 800 |
Fracture energy (\(\mathrm {N/m}\)) | G | 22, 170 |
Kalthoff’s test: number of degrees of freedom and elements corresponding to each sub-domain, for the adopted domain decompositions (see Fig. 12b)
Degrees of freedom | Elements | |||
---|---|---|---|---|
DD(2sd)\(_a\)-POD | 67,785 | 243,363 | 14,151 | 52,486 |
DD(2sd)\(_b\)-POD | 121,953 | 188,808 | 25,845 | 40,792 |
In this case a speed \(V_0=16.5\) \(\frac{\text {m}}{\text {s}}\) is considered; at this low impact velocities, brittle fracture occurs with a crack propagation at an angle of about \({70^{\circ }}\). Conversely, if the impact velocity increases, a transition in the failure mode is experimentally observed: the crack propagation is governed by the formation of shear bands ahead of the notch at a negative angle of about \({10^{\circ }}\).
The numerical simulation of Kalthoff’s experiment has been discussed in several works in the finite element literature. In [31] the problem of brittle failure was handled by applying the extended finite element technique (XFEM) on a 2-D finite element model, both with loss of hyperbolicity criterion and tensile stress criterion: the authors reported a crack propagation angle almost equal to \({58^{\circ }}\) in the former case and to \({65^{\circ }}\) in the latter one. These results have been compared by the authors with those deriving from simulations performed with inter-element technique, modelled with the Xu–Needleman’s cohesive law [24], in which the fracture propagated with an average angle of almost \({55^{\circ }}\). The XFEM method was adopted also by Combescure and co-workers in [55, 56], obtaining a crack propagation angle of \({65^{\circ }}\).
Because of the twofold symmetry, only one half of the specimen is modelled. An average elements size of 1 mm, smaller than the cohesive length equal to 6.58 mm, is considered. The resulting finite element mesh, characterized by 66, 637 quadratic tetrahedral elements and 102, 404 nodes, is shown in Fig. 12a. Symmetric boundary conditions are imposed at the lower surface of the domain.
Figure 12b shows the considered sub-divisions into two sub-domains. In this example, the domain is partitioned in such a way that the crack propagates only in one sub-domain. Table 6 gathers the number of degrees of freedom and elements corresponding to such sub-domain decompositions.
The duration of the training phase is set equal to 0.01 ms, within which 400 snapshots are collected for each sub-domain.
Figures 13, 14 and 15 show the time evolution of the crack pattern on the contour plot of \(\sigma _x\) component of the stress vector. A deviation in the crack direction with an angle almost equal to \({70^{\circ }}\) can be observed; the crack does not propagate along a single straight line, because it is constrained to follow the finite element mesh. The results obtained with the two reduced-order simulations are almost indistinguishable from the ones obtained with the DD reference algorithm.
Similarly to the results of the previous example, Table 7 shows that the coupled use of DD and POD allows to obtain a computational gain up to 25% with respect to the DD reference solution. Table 7 shows also that when a partition allows to optimize the dimension of the sub-domain which remains elastic (case \((2sd)_b\)), the computational gain is higher than in the other case (\((2sd)_a\)), even if the total number of the POMs is similar.
Kalthoff’s test: run time, computational gain with respect to the DD approach, and number of POMs retained in the analyses
Run time (s) | Gain w.r.t. DD (%) | POMs | |
---|---|---|---|
DD(2sd)\(_a\)-POD | 62, 978 | \({-}12.3\) | 78 |
DD(2sd)\(_b\)-POD | 54, 364 | \({-}24.2\) | 67 |
Conclusion
In this paper we have proposed a combination of domain decomposition and proper orthogonal decomposition strategies for the efficient simulation of fracture processes in quasi-brittle materials. The obtained results confirm the advantages of the proposed methodology and are extremely encouraging in view of a full exploitation of parallel computing.
To ensure stability of the solution in terms of inter-domain continuity, the interface phase of the domain decomposition approach has been computed through the full-order model, even after the end of the training stage of the reduced-order models. The results have shown that the computational gain can thus depend much on the partitioning of the whole body into sub-domains. As the propagation path of a main crack, or the pattern of cohesive micro-cracking cannot be foreseen under general (mixed-mode) loading conditions, the optimal design of the domain decomposition can be hardly attained.
Work in progress includes the use of strategies similar to the one here proposed for the simulation of irreversible phenomena in the presence of multi-physics coupling.
Declarations
Author's contributions
Authors contributed equally to this work. All authors read and approved the final manuscript.
Acknowledgements
SM gratefully acknowledges the financial support of Fondazione Cariplo through project Safer Helmets, Grant 2013-0716.
Competing interests
The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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