In-situ adaptive reduction of nonlinear multiscale structural dynamics models

He, Wei; Avery, Philip; Farhat, Charbel

doi:10.48550/arxiv.2004.00153

Cited by 3 publications

(5 citation statements)

References 23 publications

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“…However, the generation of training data can be a substantial cost. We propose an application of model-order reduction (see Appendix A) using a novel in-situ training approach to accelerate this task; this will be the topic of a companion paper 43 .…”

Section: Discussionmentioning

confidence: 99%

A Computationally Tractable Framework for Nonlinear Dynamic Multiscale Modeling of Membrane Fabric

Avery,

Huang,

et al. 2020

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A general-purpose computational homogenization framework is proposed for the nonlinear dynamic analysis of membranes exhibiting complex microscale and/or mesoscale heterogeneity characterized by in-plane periodicity that cannot be effectively treated by a conventional method, such as woven fabrics. The proposed framework is a generalization of the "Finite Element squared" (or FE 2 ) method in which a localized portion of the periodic subscale structure -typically referred to as a Representative Volume Element (RVE) -is modeled using finite elements. The numerical solution of displacement-driven problems using this model furnishes a mapping between the deformation gradient and the first Piola-Kirchhoff stress tensor. This unconventional material model can be readily applied in the context of membranes by using a variant of the approach proposed by Klinkel and Govindjee 1 for using conventional, finite strain, three-dimensional material models in beam and shell elements. The approach involves the numerical enforcement of the plane stress constraint, which is typically performed on out-of-plane components of the second Piola-Kirchhoff stress tensor ( ). Observing the principal of frame invariance, the RVE solution is reinterpreted as a mapping between the right stretch tensor and the symmetric Biot stress tensor conjugate pair. This facilitates the development of a drop-in replacement for any conventional finite strain plane stress material model formulated in terms of the in-plane components of the Green-Lagrange strain tensor and . Finally, computational tractability is achieved by introducing a regression-based surrogate model to avoid further solution of the RVE model when data sufficient to fit a model capable of delivering adequate approximations is available. For this purpose, a physics-inspired training regimen involving the utilization of our generalized FE 2 method to simulate a variety of numerical experiments -including but not limited to uniaxial, biaxial and shear straining of a material coupon -is proposed as a practical method for data collection. The proposed framework is demonstrated for a Mars landing application involving the supersonic inflation of an atmospheric aerodynamic decelerator system that includes a parachute canopy made of a woven fabric. Several alternative surrogate models are evaluated including a neural network.

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Section: Discussionmentioning

confidence: 99%

A Computationally Tractable Framework for Nonlinear Dynamic Multiscale Modeling of Membrane Fabric

Avery,

Huang,

et al. 2020

Preprint

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“…Figure 3 depicts these meshes, while Table 3 reports the corresponding parameters. Figure 4 plots the FOM reference solutions on the "fine" mesh with two different parameter values µ = (1, 1) and µ = (10,10). Applying these finite-element discretizations to Eq.…”

Section: Parameterized Heat Equationmentioning

confidence: 99%

“…Besides the RBE family mentioned above, researchers have developed other DDROM methods to solve parameterized linear PDEs in the context of multiscale heterogeneous materials analysis. These methods include the multiscale reduced basis method (MsRBM) [9], FE 2 -based model order reduction method [10], the localized reduced basis multiscale method (LRBMS) [11,12], the reduced basis localized orthogonal decomposition method (RB-LOD) [13], the reduced basis method for heterogeneous domain decomposition (RBHDD) [14] and recently the ArbiLoMod method [15]. In addition, we are also aware of the use of DDROM in the work of graphic community, for example (not a comprehensive list), [16,17] deal with nonlinear problems while [18,19] handle linear problems.…”

Section: Introductionmentioning

confidence: 99%

“…While some DDROM techniques have been applied to nonlinear PDEs, most of these techniques are multiscale in nature, meaning that they apply a ROM to only a subset of the physical domain, and apply the high-fidelity model elsewhere; compatibility between the ROM and high-fidelity-model solutions is enforced using non-overlapping domain decomposition methods and some multiscale homogenization assumptions [10]. For example, in the work by Buffoni and coworkers [21], the authors implemented the overlapping classical Schwarz method (using Dirichlet-Neumann iterations [22]) and divided the computational domain into two subdomains.…”

Section: Introductionmentioning

confidence: 99%

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Domain-decomposition least-squares Petrov-Galerkin (DD-LSPG) nonlinear model reduction

Hoang,

Choi,

Carlberg

2020

Preprint

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While reduced-order models have demonstrated success in many applications across computational science and engineering, they encounter challenges when applied both to truly extreme-scale models due to the prohibitive cost of generating requisite training data, and to decomposable systems due to many-query problems (e.g., design) often requiring repeated reconfigurations of system components. To address these challenges, we propose the domain-decomposition least-squares Petrov-Galerkin (DD-LSPG) model-reduction method applicable to parameterized systems of nonlinear algebraic equations (e.g., arising from discretizing a parameterized partial-differential-equations problem). In contrast with previous works, we adopt an algebraically non-overlapping decomposition strategy rather than a spatialdecomposition strategy, which facilitates application to different spatial-discretization schemes. Rather than constructing a low-dimensional subspace for the entire state space in a monolithic fashion-which would be infeasible for extreme-scale systems and decomposable models-the methodology constructs separate subspaces for the different subdomains/components characterizing the original model. During the offline stage, the method constructs low-dimensional bases for the interior and interface of subdomains/components. During the online stage, the approach constructs an LSPG reduced-order model for each subdomain/component (equipped with hyper-reduction in the case of nonlinear operators), and enforces strong or weak compatibility on the 'ports' connecting them. We propose several different strategies for defining the ingredients characterizing the methodology: (i) four different ways to construct reduced bases on the interface/ports of subdomains, and (ii) different ways to enforce compatibility across connecting ports. In particular, we show that the appropriate compatibility-constraint strategy depends strongly on the basis choice. In addition, we derive a posteriori and a priori error bounds for the DD-LSPG solutions. Numerical results performed on nonlinear benchmark problems in heat transfer and fluid dynamics that employ both finite-element and finite-difference spatial discretizations demonstrate that the proposed method performs well in terms of both accuracy and (parallel) computational cost, with different choices of basis and compatibility constraints yielding different performance profiles.

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“…All of them rely on partitioning the collected set of N s training snapshots S = {u (s) } Ns s=1 , where u (s) = u m (µ q ) is a discrete approximation of u(t m ; µ q ), t m ∈ [0, T f ], and µ q ∈ P. However, these methods differ by how they specifically partition S into subsets of solution snapshots. For example, snapshot partitioning has been performed by simply partitioning the time [34] or parameter [35,36] domain. Alternatively, state space (or HDM-based solution manifold) partitioning has been advocated and realized by clustering and compressing the solution snapshots [32]: this enables the construction of local, nonlinear PROMs capable of capturing the different regimes and features (e.g., discontinuities and fronts) that may be experienced by the solution of an HDM such as (1) [32], as well as capturing the effects on this solution of variations in the parameters of such an HDM [37].…”

Section: Construction Of a Piecewise-affine Local Subspace Of Approxi...mentioning

confidence: 99%

Mesh sampling and weighting for the hyperreduction of nonlinear Petrov-Galerkin reduced-order models with local reduced-order bases

Grimberg¹,

Farhat²,

Tezaur³

et al. 2020

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The energy-conserving sampling and weighting (ECSW) method is a hyperreduction method originally developed for accelerating the performance of Galerkin projection-based reduced-order models (PROMs) associated with large-scale finite element models, when the underlying projected operators need to be frequently recomputed as in parametric and/or nonlinear problems. In this paper, this hyperreduction method is extended to Petrov-Galerkin PROMs where the underlying high-dimensional models can be associated with arbitrary finite element, finite volume, and finite difference semi-discretization methods. Its scope is also extended to cover local PROMs based on piecewise-affine approximation subspaces, such as those designed for mitigating the Kolmogorov n-width barrier issue associated with convection-dominated flow problems. The resulting ECSW method is shown in this paper to be robust and accurate. In particular, its offline phase is shown to be fast and parallelizable, and the potential of its online phase for large-scale applications of industrial relevance is demonstrated for turbulent flow problems with O(10 7 ) and O(10 8 ) degrees of freedom. For such problems, the online part of the ECSW method proposed in this paper for Petrov-Galerkin PROMs is shown to enable wall-clock time and CPU time speedup factors of several orders of magnitude while delivering exceptional accuracy.

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In-situ adaptive reduction of nonlinear multiscale structural dynamics models

Abstract: Conventional offline training of reduced-order bases in a predetermined region of a parameter space leads to parametric reduced-order models that are vulnerable to extrapolation. This vulnerability manifests itself whenever a queried

Cited by 3 publications

References 23 publications

A Computationally Tractable Framework for Nonlinear Dynamic Multiscale Modeling of Membrane Fabric

A Computationally Tractable Framework for Nonlinear Dynamic Multiscale Modeling of Membrane Fabric

Domain-decomposition least-squares Petrov-Galerkin (DD-LSPG) nonlinear model reduction

Mesh sampling and weighting for the hyperreduction of nonlinear Petrov-Galerkin reduced-order models with local reduced-order bases

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