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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).
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. Despite ever increasing Computational resources, dissolving the fine scales in a direct numerical simulation is prohibitive. This work aims to develop an efficient approach to modeling nonlinear heterogeneous structures using the variational multiscale method (VMM) and model order reduction (MOR).
The VMM, introduced in, assumes an additive split of the solution into coarse and fine scale contributions. In, the VMM is applied to a damage mechanics–based material model for concrete-like materials. Herein, suitable boundary conditions for the fine scale which enable localization phenomena to evolve are discussed. As such, zero jump conditions between fine scale solutions are proposed which are enforced pointwise by a Lagrange type method leading to a coupled solution procedure.
In this contribution, possible extensions of the VMM with reduced order modeling are presented. In the linear case, assuming the fine scale solution to be zero on coarse scale element boundaries allows for static condensation and a decoupled solution procedure. Based on this, an efficient localized Training strategy will be developed. For the nonlinear case, the situation of coupled non-conforming spaces, i. e. finite element and reduced order spaces for the fine scales, arises. Thus the imposition of suitable fine scale interface conditions in the weak sense by the use of Lagrange multipliers is investigated. Specific problems in solid mechanics are used to illustrate the performance of the above approaches.
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).
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).
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.
In this contribution, a methodology for fine scale modeling of large scale structures is proposed, which combines the variational multiscale method[1], domain decomposition and model order reduction. The influence of the fine scale on the coarse scale is modelled by the use of an additive split of the displacement field, addressing applications without a clear scale separation. Based on the work of Buhr and Smetana[2], local reduced spaces are constructed by solving an oversampling problem with random boundary conditions. Herein, we inform the boundary conditions by a global reduced problem and compare our approach using physically meaningful correlated samples with existing approaches using uncorrelated samples. The local spaces are designed such that the local contribution of each subdomain can be coupled in a conforming way, which also preserves the sparsity pattern of standard finite element assembly procedures. Several numerical experiments show the accuracy and efficiency of the method, as well as its potential to reduce the size of the local spaces and the number of training samples compared to the uncorrelated sampling.
Multiscale modeling of linear elastic heterogeneous structures via localized model order reduction
(2023)
In this paper, a methodology for fine scale modeling of large scale linear elastic structures is proposed, which combines the variational multiscale method, domain decomposition and model order reduction. The influence of the fine scale on the coarse scale is modelled by the use of an additive split of the displacement field, addressing applications without a clear scale separation. Local reduced spaces are constructed by solving an oversampling problem with random boundary conditions. Herein, we inform the boundary conditions by a global reduced problem and compare our approach using physically meaningful correlated samples with existing approaches using uncorrelated samples. The local spaces are designed such that the local contribution of each subdomain can be coupled in a conforming way, which also preserves the sparsity pattern of standard finite element assembly procedures. Several numerical experiments show the accuracy and efficiency of the method, as well as its potential to reduce the size of the local spaces and the number of training samples compared to the uncorrelated sampling.
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. Thus, it is necessary to take into account the heterogeneous structure on the smaller scale, to capture the underlying physical phenomena. However, this poses a great challenge to the numerical solution since the computational cost is significantly increased by resolving the smaller scale in the model. Moreover, in applications where scale separation as the basis of classical homogenization schemes does not hold, the influence of the smaller scale on the larger 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) and model order reduction techniques. First, the influence of the smaller scale on the larger 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 smaller scale, is introduced. Second, local reduced approximation spaces for the smaller scale solution are constructed by exploring possible solutions for each subdomain based on the concept of oversampling and the solution of the associated transfer operator problem. Herein, we propose to choose the training data based on the solution of a reduced global problem to incorporate the actual physical behaviour of the structure of interest and to extend it by random samples to ensure sufficient approximation capabilities in general. 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).
In the field of computational science and engineering, workflows often entail the application of various software, for instance, for simulation or pre- and postprocessing.
Typically, these components have to be combined in arbitrarily complex workflows to address a specific research question. In order for peer researchers to understand, reproduce and (re)use the findings of a scientific publication, several challenges have to be addressed. For instance, the employed workflow has to be automated and information on all used software must be available for a reproduction of the results. Moreover, the results must be traceable and the workflow documented and readable to allow for external verification and greater trust.
In this paper, existing workflow management systems (WfMSs) are discussed regarding their suitability for describing, reproducing and reusing scientific workflows. To this end, a set of general requirements for WfMSs were deduced from user stories that we deem relevant in the domain of computational science and engineering. On the basis of an exemplary workflow implementation, publicly hosted at GitHub (https://github.com/BAMresearch/NFDI4IngScientificWorkflowRequirements), a selection of different WfMSs is compared with respect to these requirements, to support fellow scientists in identifying the WfMSs that best suit their requirements.
Software-driven scientific workflows are often characterized by a complex interplay of various pieces of software executed in a particular order. The output of a computational step may serve as input to a subsequent computation, which requires them to be processed sequentially with a proper mapping of outputs to inputs. Other computations are independent of each other and can be executed in parallel. Thus, one of the main tasks of a workflow tool is a proper and efficient scheduling of the individual processing steps.
Each processing step, just as the workflow itself, typically processes some input and produces output data. Apart from changing the input data to operate on, processing steps can usually be configured by a set of parameters to change their behavior. Moreover, the behavior of a processing step is determined by its source code and/or executable binaries/packages that are called within it. Beyond this, the computation environment not only has a significant influence on its behavior, but is also crucial in order for the processing step to work at all. The environment includes the versions of the interpreters or compilers, as well as all third-party libraries and packages that contribute to the computations carried out in a processing step.