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A hyper reduced domain decomposition approach for modeling nonlinear heterogeneous structures
(2019)
Many of today's problems in engineering demand reliable and accurate prediction of failure mechanisms of mechanical structures. Herein it is necessary to take into account the often heterogeneous structure on the fine scale, to capture the underlying physical phenomena. However, an increase of accuracy by dissolving the fine scale inevitably leads to an increase in computational cost. In the context of multiscale simulations, the FE2 method is widely used. In a two-level computation, the fine scale is depicted by a boundary value problem for a representative volume element (RVE), which is then solved in each integration point of the macro scale to determine the macroscopic response. However, the FE2 approach in general is computationally expensive and problematic in the special case of concrete structures. Here rather large RVEs are necessary to sufficiently represent the meso-structure, such that separation of scales cannot be assumed.
Therefore, the aim is to develop an efficient approach to modeling nonlinear heterogeneous structures using domain decomposition and reduced order modeling.
A safe and robust performance is a key criterion when building and maintaining structures and component. Ensuring this criterion at different stages of the lifetime can be supported by applying continuous monitoring concepts. The latter usually can serve multiple purposes, including the determination of material parameters for the design phase, the evaluation of the actual loading/environmental conditions (instead of using conservative estimates that are usually larger) and evaluating or predicting the true performance of the structure (thus decreasing the model bias). In this context, a digital twin of the structure has many benefits. It allows to introduce virtual sensors to “measure” sensor information that is e.g. inaccessible or unmeasureable. In order to efficiently use monitoring techniques in the context of a digital twin, it is important to consider the complete chain of information including the choice of sensors, the data processing and structuring, the modelling assumptions, the numerical simulation and finally the stochastic nature of the model prediction. In this presentation, challenges in this context are discussed with a specific focus on Bayesian model updating of the digital twin, accounting for both parameter updates as well as model bias that results from the limitations of modelling assumption. A bottleneck in this approach is the computational effort related to sampling methods such as Markov chain Monte Carlo methods that require many evaluations of the forward model. An alternative to the expensive computation of the forward model for updating the digital twin is the combination with model reduction techniques such as the Proper General Decomposition [1, 2]. The results are illustrated for several examples and scale, ranging from digitals twin for material tests in the lab over lab scale structural digital twins up to damage identification in field experiments.
In analyzing large scale structures, it is necessary to take into account the material heterogeneity for accurate failure prediction. However, this greatly increases the degrees of freedom in the numerical method thus making it infeasible. Moreover, in applications where scale separation as the basis of classical homogenization schemes does not hold, the influence of the fine scale on the coarse scale
has to be modelled directly.
This work aims to develop an efficient methodology to model heterogeneous structures combining the variational multiscale method and model order reduction techniques. Superposition-based methods assume a split of the solution field into coarse and fine scale contributions. In deriving practical methods, some form of localization is necessary to eliminate the fine-scale part from the coarse-scale equation. Hund and Ramm [2] discussed different locality constraints and in particular zero jump conditions enforced by a Lagrange-type method leading to a coupled solution scheme.
In this contribution, a combination of the variational multiscale method and model order reduction techniques is applied to model the influence of the fine scale on the coarse scale directly. First, possible coarse and fine scale solutions are exploited for a representative volume element (RVE), specific to the material of interest, to construct local approximation spaces. For the local fine scale spaces different choices are presented, which ensure continuity between adjacent coarse grid elements. Therefore,the resulting global system takes into account, the effect of the fine scale on the coarse scale, is sparse and has much lower dimensions compared to the full system in the direct numerical simulation.
The authors gratefully acknowledge financial support by the German Research Foundation (DFG), project number 394350870. This result is part of a project that has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No. 818473).
In analyzing large scale structures, it is necessary to take into account the material heterogeneity for accurate failure prediction. However, this greatly increases the degrees of freedom in the numerical method making it infeasible. Moreover, in applications where scale separation as the basis of classical homogenization schemes does not hold, the influence of the fine scale on the coarse scale has to be modelled directly.
This work aims to develop an efficient methodology to model heterogeneous structures combining the variational multiscale method and model order reduction techniques.
Superposition based methods assume a split of the solution field into coarse and fine scale contributions. In deriving practical methods some form of localization is necessary to eliminate the fine scale part from the coarse scale equation. Hund and Ramm discussed different locality constraints and resulting solution procedures in the context of solid mechanics. Particularly, zero jump conditions ensuring continuity of the fine scale solution which are enforced by a Lagrange type method lead to a coupled solution procedure.
In this contribution, a combination of the variational multiscale method and model order reduction techniques is applied to model the influence of the fine scale on the coarse scale directly. First, possible coarse and fine scale solutions are exploited for a representative volume element (RVE), specific to the material of interest, to construct local approximation spaces. The local spaces are designed such that local contributions of RVEs can be coupled in a conforming way. Therefore, the resulting global system takes the effect of the fine scale on the coarse scale into account, is sparse and reduced in size compared to the direct numerical simulation.
The authors gratefully acknowledge financial support by the German Research Foundation (DFG), project number 394350870, and by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (ERC Grant agreement No. 818473).
Multiscale modeling of heterogeneous structures based on a localized model order reduction approach
(2022)
Many of today’s problems in engineering demand reliable and accurate prediction of failure mechanisms of mechanical structures. Herein, it is necessary to take into account the heterogeneous structure on the lower scale, to capture the underlying physical phenomena. However, this poses a great challenge to the numerical solution as the computational cost is significantly increased by resolving the lower scale in the model. Moreover, in applications where scale separation as the basis of classical homogenization schemes does not hold, the influence of the lower scale on the upper scale has to be modelled directly. This work aims to develop an efficient concurrent methodology to model heterogeneous structures combining the variational multiscale method (VMM) [1] and model order reduction techniques (e. g. [2]). First, the influence of the lower scale on the upper scale can be taken into account following the additive split of the displacement field as in the VMM. Here, also a decomposition of the global domain into subdomains, each containing a fine grid discretization of the lower scale, is introduced. Second, reduced approximation spaces for the upper and lower scale solution are constructed by exploring possible solutions for each subdomain based on a representative unit cell. The local reduced spaces are designed such that local contributions of each subdomain can be coupled in a conforming way. Thus, the resulting global system is sparse and reduced in size compared to the direct numerical simulation, leading to a faster solution of the problem. The authors gratefully acknowledge financial support by the German Research Foundation (DFG), project number 394350870, and by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (ERC Grant agreement No. 818473).
There is a rising attention of using numerical models for effcient structural monitoring and ensuring the structure's safety. Setting up virtual models as twin for real structures requires a model identification process calculating the unknown model parameters, which mostly are only indirectly measurable. This is a computationally very costly inverse optimization process, which often makes it unfeasible for real applications. Effcient surrogate models such as reduced order models can be used, to overcome this limitation. But the influence of the model accuracy on the identification process has then to be considered. The aim is to automatically control the influence of the model's accuracy on the identification. Here, a variational Bayesian inference approach[3] is coupled with a reduced forward model using the Proper Generalized Decomposition (PGD) method. The influence of the model accuracy on the inference result is studied and measured. Therefore, besides the commonly used Bayes factor the Kullback-Leibler divergences between the predicted posterior pdfs are proposed. In an adaptive inference procedure, the surrogate's accuracy is iteratively increased, and the convergence of the posterior pdf is analysed. The proposed adaptive identification process is applied to the identification of spatially distributed damage modeled by a random eld for a simple example with synthetic data as well as a small, reinforced bridge with real measurement data. It is shown that the proposed criteria can mirror the influence of the model accuracy and can be used to automatically select a suffciently accurate surrogate model.