In this section, the results of a transient, non-linear example including contact formulations are presented.For the intrusive MOR scheme the software LS-Dyna including a special implementation interface is utilised. The implementation was done within a previous work by Bach (2020) and has been further extended to perform optimisation studies. A stand-alone Python code holds the algorithm for the non-intrusive scheme, which was developed for this study.
To assess the two algorithms, a crash box exampleFootnote 1 is introduced. The crash box, as depicted in Fig. 11, is modelled as an elasto-plastic tube with a wall thickness of 2.0 mm and a length of 272.5 mm, whereby imperfections are introduced along the tube to trigger the folding mechanism. A rigid plate with an initial velocity of 40 km/h in negative z-direction crushes the tube, which is clamped at the bottom.
All deformable parts are discretised with fully integrated Reissner-Mindlin shell elements (ELFORM=16), which have translational and rotational degrees of freedom. In contrast to the element formulation the remaining contact and material parameter have not been changed in comparison to the template\(^{1}\). Penalty formulations are applied to model the contact between plate and crash box as well as the self-contact (CONTACT_AUTOMATIC_SINGLE_SURFACE). The material properties for the crash box are as follows: The mass density is \(\rho = 7830\) kg/m\(^3\), the Young’s modulus is \(E = 200\) GPa, the Poisson’s ratio is \(\nu = 0.30\) and the yield strength is \(\sigma _y = 0.366\) GPa with a piece-wise linear plasticity model. The reduced order models are constructed for the tube discretised by 1924 nodes and a termination time of T = 20 ms is set.
For the intrusive scheme, the projection matrix \({\mathbf {V}}\) is orthonormalised with respect to the mass matrix and built up for displacement and rotational degrees of freedom separately. As the focus is on transient analysis an error measure considering the full time domain is defined. To evaluate the accuracy the global mean relative error (GMRE) is computed using the full order displacements, \({\mathbf {u}}\), and the reduced order displacements \(\mathbf {u_r}\) for the full time domain T as follows:
$$\begin{aligned} \varepsilon _{GMRE} := \frac{\sqrt{ \sum \limits _{t \in T } ({\mathbf {u}}(t) - {\mathbf {V}}\mathbf {u_{r}}(t))^{T} ({\mathbf {u}}(t) - {\mathbf {V}}\mathbf {u_{r}}(t)) } }{ \sqrt{ \sum \limits _{t \in T } {\mathbf {u}}^{T}(t) {\mathbf {u}}(t) }}. \end{aligned}$$
(16)
In the following, the intrusive and non-intrusive MOR schemes are applied to the crash box example and their results are compared regarding accuracy and numerical effort. The discussion of the results is split into training accuracy and online accuracy. The former evaluates the ability to reproduce the training data, while the latter examines the performance on parameters that are not present in the training data. Furthermore, their inter- and extrapolation capabilities are studied, whereby the impacting kinetic energy and the thickness of the crash box are varied. To evaluate their overall computational cost a comparative analysis is presented. In Sect. 6 the reduced models of the crash box are embedded into an optimisation workflow.
Training accuracy
To test the intrusive and non-intrusive scheme for transient analysis the first study evaluates a model reproducing the training simulation. Thereby, the parameter domain P is neglected and only the time domain T is considered. Uniformly, every \(t = 0.01\)ms a snapshot is allocated to the snapshot matrix.
Figure 3 shows the results of two crash box simulations computed by the intrusive MOR, whereby the grey wireframe represents the full order model (FOM) and the orange and green shells indicate the selected hyper-reduction elements for two different reduction levels (Bach 2020). The parameter k is the number of basis vectors for the Galerkin projected reduced order model (ROM), and \(\tau\) is the tolerance value for the hyper-reduction algorithm in Eq. (8).
The Galerkin ROM and the hyper-reduced model (HROM), as depicted in Fig. 3, result in a computational speed-up factors of 4.7 and 7.1, respectively.
To further validate the approach, Fig. 4 shows the displacement using intrusive and non-intrusive MOR of two reference nodes (highlighted in blue and green in Fig. 5), which are included in the folding mechanism. The intrusive reduced order model (ROM) on the left shows the nodal displacement result of the FOM and the Galerkin ROM and HROM with larger \(\Delta t\), as Galerkin projection leads to a higher critical time step for explicit solvers (Bach et al. 2018; Krysl et al. 2001). On the right of Fig. 4 the results of the reference node are plotted utilising the non-intrusive model with \(k=20\) basis vectors and a Gaussian Process regression (isotropic Matérn kernelFootnote 2). Hereby, no physical system is solved and no contact algorithm is evaluated.
This example illustrates, that the non-intrusive regression model can be fitted to the example data. To analyse the quality of non-intrusive ROMs, different regression models are fitted and tested with varying input parameter configurations in the next section.
Online accuracy
The previous section presented the ability of the MOR schemes to create a simplified model for a non-linear structural simulation. However, a replication of the training simulation is not the intention of MOR methodology as no computational efficiency is gained. Considering a multi-query analysis, such as optimisation or probabilistic analysis, the idea is to roughly identify the parameter space and perform a couple of high-fidelity simulations beforehand. These are utilised for the construction of the reduced model to enable fast online simulations within the multi-query analysis. Therefore, the inter- and extrapolation capabilities within the online phase are the key point for an efficient application. A study investigating two different non-linear manifolds should illustrate and compare the capabilities of intrusive and non-intrusive models for transient analysis. We first restrict us to a one-dimensional parameter space, whereby a larger space is evaluated within the optimisation study in the next section.
To evaluate the effect of parameter variations the kinetic energy of the impacting plate and the thickness of the crash box are identified as suitable variables. For both parameters individual models are created for a variation of \(\pm 20 \%\) for all following studies. To change the kinetic energy applied to the crash box, the mass of the impacting plate is varied by \(\pm 20 \%\), as it deviates proportionally.
The variation of kinetic energy and thickness under constant velocity has a drastic impact on the folding mechanism of the crash box depicted in Fig. 5. It is also noticeable, that the effects of the varying kinetic energy have a smaller impact on the system than \(20 \%\) deviation of thickness. By discussing both variables we can investigate the different “levels” of non-linearity and their effects on the online accuracy.
To analyse the capabilities of ROMs representing varying manifolds, first a study focusing on the non-intrusive approach is presented. The non-intrusive ROMs exploiting the machine learning (ML) techniquesFootnote 3 polynomial regression (poly), k-nearest neighbour regression (kNN) and Gaussian Process regression (GPR) are compared individually for the two cases: impacting kinetic energy and thickness of the tube. As the models highly depend on the quality of the training phase, not only the different regression techniques are assessed, but also an increasing number of training simulations.
To judge data-driven meta models, a prior distinction between training and test data enables the computation of multiple error measures such as the \(R^2\)- value or global mean relative error (GMRE). A training data set is created by Sobol samplingFootnote 4 with increasing sample set \(\mu _1,\mu _2, ..., \mu _{n_{\mu }} \in P\) with \(n_{\mu }\) of 2, 4, 8, 16, and 32. In addition, a testing set of 32 samples is built by a random Sobol sequence, whereby each number is multiplied by a scaled random value. The distribution of the corresponding training and test samples are depicted in Fig. 6, whereby the test points are highlighted with vertical blue lines.
The training and testing configurations are scaled for the parameter space thickness of the crash box \(t_{tube} \in P\) and energy of the impacting plate \(e_{impact} \in P\). Full order analyses are simulated accordingly and snapshots are collected uniformly every 0.1ms. Five ROMs \(\phi (t,t_{tube})\) are individually built for the sets of 2, 4, 8, 16, and 32 training points \(n_{\mu }\), whereby each simulation contributes 200 snapshots \(t_1,t_2,..,t_{200} \in T\). To create five ROMs \(\phi (t,e_{impact})\) the procedure is repeated. All models are based on a reduced subspace with \(k=20\) basis vectors.
In Fig. 8 the displacement GMRE of Eq. (16) is plotted, as an error measure for all 32 test samples. The bar is drawn from the smallest to the largest GMRE and the mean GMRE of all samples is highlighted individually by a marker for kNN, poly and GPR. For all models, the accuracy increases with the number of training simulations. One can notice, that the poly models of order seven, constructed from two training simulations do not provide useful surrogates, possibly due to over fitting. Despite this exception, the regression techniques kNN with five neighbours, poly and GPR using an anisotropic Matérn kernel show a GMRE in similar ranges.
Table 1 compares the GMRE and \(R^2\) for the models \(\phi _8(t,t_{tube})\) and \(\phi _8(t,e_{impact})\) built from 8 training simulations for all three ML techniques. It is noticeable, that the GPR performs best in the framework of non-intrusive MOR with a GMRE of 0.28% and \(R^2 = 0.999\) for the parameter space \(e_{impact}\). For this example, the models built by kNN have a higher accuracy than those obtained by polynomial regression. As the kNN technique averages a data point with the k-nearest neighbours, its quality to approximate the very first and last time step is reduced. The performance of kNN trained on sparse data, especially in the time domain can drop significantly, as also observed by Kneifl et al. (2021). However, kNN is a fast and robust technique and especially for high number of data points a simpler regression model can be beneficial.
Comparing \(R^2\) and GMRE for the different parameter domains, all \(\phi (t,e_{impact})\) regression models show a higher \(R^2\) than \(\phi (t,t_{tube})\). This can also be observed in Fig. 8 for models with increasing training sets. On the left \(\phi (t,t_{tube})\) models have continuously higher GMREs as \(\phi (t,e_{impact})\) surrogates. As a smaller range of displacement patterns (Fig. 5) corresponds to the variation of the impacting kinetic energy \(e_{impact}\), this could be expected.
To further understand the particular meta models, the displacement in x, y, and z direction of the first test simulation are visualised for two folding points (marked in Fig. 5) in Fig. 7. Similar to the table 1 poly, kNN, and GPR are based on snapshots from 8 transient training simulations and a subspace of \(k=20\) basis vectors. An artificially smooth function can be observed for the polynomial regression of order seven. Compared to the other techniques, GPR using an anisotropic Matérn kernel has superior accuracy (Fig. 8) and is able to depict more irregular data points, as also exploited by Guo and Hesthaven (2019).
Table 1 Comparison of mean error measures \(R^2\) and GMRE for varying ROMS with a subspace \(k=20\) and poly (p = 7), kNN (k = 5), GPR (anisotropic Matérn kernel) for 8 training simulations and 32 test simulations The next study compares the non-intrusive to the intrusive approach for varying parameter domains. For the intrusive MOR the question rises if the projected system of equation enables an extrapolation of design variables. Therefore, the projection matrix is constructed from a single training simulation and tested in the online phase with extrapolating design variables.
The displacement GMRE by Eq. (16) of the intrusive MOR are plotted on the right of Fig. 9a and b. The model is trained, collecting snapshots every 0.01ms, using a plate mass of 150 kg and a wall thickness of 2.0 mm. It can be observed, that the error increases with the distance to the training configuration. Also the relative error due to a change in wall thickness is generally higher than the error associated with a change of the plate’s mass. This further supports the observation of higher nonlinearities associated with a varying tube thickness.
In contrast to the small extrapolation capabilities of the intrusive ROMs, the non-intrusive scheme is restricted to interpolation. Training a non-intrusive model with snapshots corresponding to one parameter instance would yield unreasonable results, as the surrogate is essentially a regression model and only enriched by physical phenomena. For a comparison of the techniques, the minimum number of training simulations to reach a similar accuracy level is investigated. The snapshot time increments are enlarged to 1ms, such that a total number of 20 snapshots are collected from each transient analysis. The number of training simulations was successively increased until the non-intrusive ROMs achieved GMREs in the same range as the intrusive approach. As for non-intrusive models the computational cost mainly depends on the number of training simulations; this is an import factor to compare the overall efficiency gains.
On the left of Fig. 9a and b the displacement GMRE of the non-intrusive MOR results are depicted. The training samples are obtained by Latin hypercube sampling and marked with grey circles in Fig. 9a and b. Hence, three training simulations are needed for varying mass and thirteen for varying thickness, in order to obtain similar error values compared to the intrusive MOR method. As already observed before, the accuracy of the simplified model strongly depends on the non-linear manifold and the quality of the training phase. Note, that for the variation of thickness in Fig. 9b the number of training data rises to thirteen, which would be equivalent to an increasing online error while keeping the same number of training simulations.
It could be shown, that non-intrusive ROMs are capable of representing the crash box example, whereby especially kNN and GPR provided reliable regression models. The GPR is the most accurate technique and kNN is favorable in terms of efficiency and robustness. Moreover, the accuracy of ROMs highly relies on the amount of training simulations and snapshot intervals in time and parameter domain. Note, that the characteristic change in time and parameter space defines the non-linear manifold and therewith the required training set. A comparison to the intrusive approach illustrates, that similar accuracy can be achieved by non-intrusive ROMs with a higher number of training data. For the intrusive ROMs of the crash box, extrapolation within the parameter domain is possible for a \(20 \%\) range, however a general conclusion cannot be drawn from this example analysis. Further studies in the field of crashworthiness are required e.g. including error measures.
Computational cost
After examining the accuracy of the models, we now focus on the computational speedup of the reduced models. The construction cost of the machine learning model is negligibly small compared to the evaluation of the training simulations for all techniques. One ROM evaluation using kNN or poly is 4 orders of magnitude and for GPR 3 orders of magnitude smaller than a full order analysis. However, for a higher number of DOFs, but especially for a higher number of design parameters and snapshots, the cost of constructing a Gaussian Process can significantly increase. For a detailed comparison of online and offline costs for the different ML techniques it is referred to Kneifl et al. (2021).
Here, we focus on the comparison of costs of the intrusive and non-intrusive method. Table 2 shows the elapsed time for the crashbox example of Fig. 9. The single online simulations of the corresponding ROMs are measured on a Intel Xeon 3.5 GHz processor with 4 CPUs. The intrusive scheme has a speed-up factor of approximately 4.7 and the non-intrusive scheme with GPR is of 3 magnitudes faster than the FOM. However, including the offline phase into the evaluation of the computational cost the results appear to be different. Figure 10 compares the cost function of intrusive and non-intrusive with \(\phi (t,e_{impact})\) and \(\phi (t,t_{tube})\), in grey, blue and green respectively. The x-axis shows the number of online evaluations, whereby the start represents the training effort. The values on the y-axis are normalised by the computational cost of one full order simulation. Thus, the intrusive training cost equals to 1 and the non-intrusive to 3 and 13. With a speed-up factor of four and \(10^4\) the efficiency of the non-intrusive scheme overtakes the performance of the intrusive for 10 and 57 online evaluations of \(\phi (t,e_{impact})\) and \(\phi (t,t_{tube})\) respectively. Note, that the costs of SVD and hyper-reduction are neglected here. The interested reader is referred to a detailed evaluation by Bach (2020).
Table 2 Elapsed time for the crashbox example of Fig. 9 comparing full order simulation (FOM), intrusive (Galerkin ROM and HROM) (Bach 2020) and non-intrusive ROM