Reduced order modeling of nonlinear microstructures through Proper Orthogonal Decomposition

Gobat, Giorgio; Opreni, Andrea; Fresca, Stefania; Manzoni, Andrea; Frangi, Attilio

doi:10.1016/j.ymssp.2022.108864

Cited by 26 publications

(29 citation statements)

References 58 publications

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“…In the case of parametrized PDEs featuring nonaffine dependence on the parameter and/or nonlinear (highorder polynomial or nonpolynomial) dependence on the field variable, a further level of reduction, known as hyper-reduction, must be introduced [46,18]. Note that if nonlinearities only include quadratic (or, at most, cubic) terms and do not feature any parameter dependence, assembling of nonlinear terms in the ROM can be performed by projection of the corresponding FOM quantities, once and for all [47].…”

Section: Hyper-reduction: the Discrete Empirical Interpolation Methodsmentioning

confidence: 99%

Deep-HyROMnet: A deep learning-based operator approximation for hyper-reduction of nonlinear parametrized PDEs

Cicci¹,

Fresca²,

Manzoni³

2022

Preprint

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To speed-up the solution to parametrized differential problems, reduced order models (ROMs) have been developed over the years, including projection-based ROMs such as the reduced-basis (RB) method, deep learning-based ROMs, as well as surrogate models obtained via a machine learning approach. Thanks to its physics-based structure, ensured by the use of a Galerkin projection of the full order model (FOM) onto a linear low-dimensional subspace, RB methods yield approximations that fulfill the physical problem at hand. However, to make the assembling of a ROM independent of the FOM dimension, intrusive and expensive hyper-reduction stages are usually required, such as the discrete empirical interpolation method (DEIM), thus making this strategy less feasible for problems characterized by (high-order polynomial or nonpolynomial) nonlinearities. To overcome this bottleneck, we propose a novel strategy for learning nonlinear ROM operators using deep neural networks (DNNs). The resulting hyper-reduced order model enhanced by deep neural networks, to which we refer to as Deep-HyROMnet, is then a physics-based model, still relying on the RB method approach, however employing a DNN architecture to approximate reduced residual vectors and Jacobian matrices once a Galerkin projection has been performed. Numerical results dealing with fast simulations in nonlinear structural mechanics show that Deep-HyROMnets are orders of magnitude faster than POD-Galerkin-DEIM ROMs, keeping the same level of accuracy.

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Section: Hyper-reduction: the Discrete Empirical Interpolation Methodsmentioning

confidence: 99%

Deep-HyROMnet: A deep learning-based operator approximation for hyper-reduction of nonlinear parametrized PDEs

Cicci¹,

Fresca²,

Manzoni³

2022

Preprint

Self Cite

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“…The POD-G approach has been recently benchmarked on several MEMS including beams, arches and mirrors. With reference to MEMS structures Gobat et al [25] have shown that, provided that a sufficient number of POD modes is included in the trial space, the technique accurately reproduces the response of the device. However, in particular when large rotations are involved and nonlinearities are strong, the dimension of the trial space increases and solution of the ROM with continuation techniques comes with a high computational cost, failing to serve the final goal of generating a real-time simulation tool.…”

Section: Pod-galerkin Rom (Pod-g Rom)mentioning

confidence: 99%

“…The generated high-fidelity snapshots are processed with SVD and a subset of POD modes is selected. The number of POD modes kept in the subspace must be defined in order to guarantee a good approximation of the underlying manifold as shown by Gobat et al [25], where POD-G ROMs are applied to mechanical systems with low damping like MEMS. By exploiting the generated linear subspace, the POD-G ROM is built following the procedure detailed in Section 3.1.…”

Section: Pod-galerkin-enhanced Deep Learning Based-reduced Order Mode...mentioning

confidence: 99%

“…Thus, the projection of dynamical systems onto low-dimensional nonlinear manifolds is still an open challenge. Data-driven approaches based on Proper Orthogonal Decomposition (POD) have been recently successfully applied to MEMS by Gobat et al [25]; however, these are still linear approaches that might experience some difficulties in reproducing the correct curvature of invariant manifolds while retaining only few DOFs in the ROM, and eventually fail to reach real-time performances.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper we propose a new version of the DL-ROM discussed above, in which a first dimensionality reduction of the data is achieved by means of a POD-Galerkin (POD-G) ROM which has been recently applied to microstructures by Gobat et al [25] showing very good predictive capabilities. In this way, the costly FOM data generation phase is greatly reduced and the training of the DL-ROM is performed using cheaper POD-G snapshots covering the whole parameter range.…”

Section: Introductionmentioning

confidence: 99%

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Deep learning-based reduced order models for the real-time simulation of the nonlinear dynamics of microstructures

Fresca¹,

Gobat²,

Fedeli³

et al. 2021

Preprint

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We propose a non-intrusive Deep Learning-based Reduced Order Model (DL-ROM) capable of capturing the complex dynamics of mechanical systems showing inertia and geometric nonlinearities. In the first phase, a limited number of high fidelity snapshots are used to generate a POD-Galerkin ROM which is subsequently exploited to generate the data, covering the whole parameter range, used in the training phase of the DL-ROM. A convolutional autoencoder is employed to map the system response onto a lowdimensional representation and, in parallel, to model the reduced nonlinear trial manifold. The system dynamics on the manifold is described by means of a deep feedforward neural network that is trained together with the autoencoder. The strategy is benchmarked against high fidelity solutions on a clamped-clamped beam and on a real micromirror with softening response and multiplicity of solutions. By comparing the different computational costs, we discuss the impressive gain in performance and show that the DL-ROM truly represents a real-time tool which can be profitably and efficiently employed in complex system-level simulation procedures for design and optimisation purposes.

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Deep learning‐based reduced order models for the real‐time simulation of the nonlinear dynamics of microstructures

Fresca

Gobat

Fedeli

et al. 2022

Numerical Meth Engineering

Self Cite

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We propose a non‐intrusive deep learning‐based reduced order model (DL‐ROM) capable of capturing the complex dynamics of mechanical systems showing inertia and geometric nonlinearities. In the first phase, a limited number of high fidelity snapshots are used to generate a POD‐Galerkin ROM which is subsequently exploited to generate the data, covering the whole parameter range, used in the training phase of the DL‐ROM. A convolutional autoencoder is employed to map the system response onto a low‐dimensional representation and, in parallel, to model the reduced nonlinear trial manifold. The system dynamics on the manifold is described by means of a deep feedforward neural network that is trained together with the autoencoder. The strategy is benchmarked against high fidelity solutions on a clamped‐clamped beam and on a real micromirror with softening response and multiplicity of solutions. By comparing the different computational costs, we discuss the impressive gain in performance and show that the DL‐ROM truly represents a real‐time tool which can be profitably and efficiently employed in complex system‐level simulation procedures for design and optimization purposes.

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Reduced order modeling of nonlinear microstructures through Proper Orthogonal Decomposition

Cited by 26 publications

References 58 publications

Deep-HyROMnet: A deep learning-based operator approximation for hyper-reduction of nonlinear parametrized PDEs

Deep-HyROMnet: A deep learning-based operator approximation for hyper-reduction of nonlinear parametrized PDEs

Deep learning-based reduced order models for the real-time simulation of the nonlinear dynamics of microstructures

Deep learning‐based reduced order models for the real‐time simulation of the nonlinear dynamics of microstructures

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