Data-Driven Filtered Reduced Order Modeling of Fluid Flows

Xie, Xuping; Mohebujjaman, Muhammad; Rebholz, Leo G.; Iliescu, Traian

doi:10.1137/17m1145136

Cited by 141 publications

(179 citation statements)

References 77 publications

(221 reference statements)

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“…Although the CDDC‐ROM and DDC‐ROM are more accurate than the G‐ROM, the computational cost of calculating

\overset{A}{true}

and

\overset{B}{true}

in the off‐line phase can be significant. Thus, in the work of Xie et al, we proposed the following practical approach for reducing the computational cost of

\overset{A}{true}

and

\overset{B}{true}

calculation: since d , the rank of the snapshot matrix, can be large in practical applications, instead of using

u_{d} \in X^{d}

to compute the Correction term in , we utilized the following approximation:

- ((), {\overset{false(u_{d} \cdot \nabla false) u_{d}}{true}}^{r} - false(u_{r} \cdot \nabla false) u_{r}, φ_{i}) \approx - ((), {\overset{false(u_{m} \cdot \nabla false) u_{m}}{true}}^{r} - false(u_{r} \cdot \nabla false) u_{r}, φ_{i}),

∀ i = 1,…, r . In , we replaced u d with u m and the ROM projection on X d with the ROM projection on X m .…”

Section: Numerical Resultsmentioning

confidence: 99%

“…For a fixed r ≤ d and a given u ∈ X h (where X h is the FE space), the ROM projection seeks

{\overset{u}{true}}^{r} \in X^{r}

such that

((), {\overset{u}{true}}^{r}, φ_{j}) = false(bold-italicu, φ_{j} false) \forall j = 1, …, r .

Filtering the NSE (see Section 3.2 in the work of Xie et al for details), we obtain the spatially filtered ROM , as follows:

((), \frac{\partial u_{r}}{\partial t}, φ_{i}) + R e^{- 1} false(\nabla u_{r}, \nabla φ_{i} false) + ((), false(u_{r} \cdot \nabla false) u_{r}, φ_{i}) + ((), τ_{r}^{SFS}, φ_{i}) = bold0,

where the ROM stress tensor is

{bold-italicτ}_{r}^{SFS} = {\overset{false(u_{d} \cdot \nabla false) u_{d}}{true}}^{r} - false(u_{r} \cdot \nabla false) u_{r} .

The spatially filtered ROM can be written as

\overset{a}{true} = A bold-italica + a^{⊤} B bold-italica + bold-italicτ,

where A and B are the same as those in and the components of τ are given by

…”

Section: Data‐driven Correction Rommentioning

confidence: 99%

“…Although the DDC‐ROM yielded good results, these results got worse when the ROM dimension was decreased below a certain threshold. To address this issue, we propose the physically constrained DDC‐ROM (CDDC‐ROM) , which aims at improving the physical accuracy of the Correction term in the DDC‐ROM.…”

Section: Introductionmentioning

confidence: 97%

“…In the work of Xie et al, we investigated the DDC‐ROM in the numerical simulation of a two‐dimensional (2D) channel flow past a circular cylinder at Reynolds numbers R e = 100, R e = 500, and R e = 1000. The DDC‐ROM was significantly more accurate than the standard Proj‐ROM.…”

Section: Introductionmentioning

confidence: 99%

“…The rest of this paper is organized as follows. In Section 2, we review the DDC‐ROM proposed in the work of Xie et al In Section 3, we propose the new CDDC‐ROM. In Section 4, we perform a numerical investigation of the new CDDC‐ROM in the numerical simulation of a 2D flow past a circular cylinder at Reynolds numbers R e = 100, R e = 500, and R e = 1000.…”

Section: Introductionmentioning

confidence: 99%

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Physically constrained data‐driven correction for reduced‐order modeling of fluid flows

Mohebujjaman

Rebholz

Iliescu

2018

Numerical Methods in Fluids

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Summary We have recently proposed a data‐driven correction reduced‐order model (DDC‐ROM) framework for the numerical simulation of fluid flows, which can be formally written as follows. The new DDC‐ROM was constructed by using reduced‐order model spatial filtering and data‐driven reduced‐order model closure modeling (for the Correction term) and was successfully tested in the numerical simulation of a two‐dimensional channel flow past a circular cylinder at Reynolds numbers Re = 100,Re = 500, and Re = 1000. In this paper, we propose a physically constrained DDC‐ROM (CDDC‐ROM), which aims at improving the physical accuracy of the DDC‐ROM. The new physical constraints require that the CDDC‐ROM operators satisfy the same type of physical laws (ie, the Correction term's linear component should be dissipative, and the Correction term's nonlinear component should conserve energy) as those satisfied by the fluid flow equations. To implement these physical constraints, in the data‐driven modeling step, we replace the unconstrained least squares problem with a constrained least squares problem. We perform a numerical investigation of the new CDDC‐ROM and standard DDC‐ROM for a two‐dimensional channel flow past a circular cylinder at Reynolds numbers Re = 100,Re = 500, and Re = 1000. To this end, we consider a reproductive regime as well as a predictive (ie, cross‐validation) regime in which we use as little as 50% of the original training data. The numerical investigation clearly shows that the new CDDC‐ROM is significantly more accurate than the DDC‐ROM in both regimes.

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“…Although the CDDC‐ROM and DDC‐ROM are more accurate than the G‐ROM, the computational cost of calculating

\overset{A}{true}

and

\overset{B}{true}

in the off‐line phase can be significant. Thus, in the work of Xie et al, we proposed the following practical approach for reducing the computational cost of

\overset{A}{true}

and

\overset{B}{true}

calculation: since d , the rank of the snapshot matrix, can be large in practical applications, instead of using

u_{d} \in X^{d}

to compute the Correction term in , we utilized the following approximation:

- ((), {\overset{false(u_{d} \cdot \nabla false) u_{d}}{true}}^{r} - false(u_{r} \cdot \nabla false) u_{r}, φ_{i}) \approx - ((), {\overset{false(u_{m} \cdot \nabla false) u_{m}}{true}}^{r} - false(u_{r} \cdot \nabla false) u_{r}, φ_{i}),

∀ i = 1,…, r . In , we replaced u d with u m and the ROM projection on X d with the ROM projection on X m .…”

Section: Numerical Resultsmentioning

confidence: 99%

“…For a fixed r ≤ d and a given u ∈ X h (where X h is the FE space), the ROM projection seeks

{\overset{u}{true}}^{r} \in X^{r}

such that

((), {\overset{u}{true}}^{r}, φ_{j}) = false(bold-italicu, φ_{j} false) \forall j = 1, …, r .

Filtering the NSE (see Section 3.2 in the work of Xie et al for details), we obtain the spatially filtered ROM , as follows:

((), \frac{\partial u_{r}}{\partial t}, φ_{i}) + R e^{- 1} false(\nabla u_{r}, \nabla φ_{i} false) + ((), false(u_{r} \cdot \nabla false) u_{r}, φ_{i}) + ((), τ_{r}^{SFS}, φ_{i}) = bold0,

where the ROM stress tensor is

{bold-italicτ}_{r}^{SFS} = {\overset{false(u_{d} \cdot \nabla false) u_{d}}{true}}^{r} - false(u_{r} \cdot \nabla false) u_{r} .

The spatially filtered ROM can be written as

\overset{a}{true} = A bold-italica + a^{⊤} B bold-italica + bold-italicτ,

where A and B are the same as those in and the components of τ are given by

…”

Section: Data‐driven Correction Rommentioning

confidence: 99%

Section: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Physically constrained data‐driven correction for reduced‐order modeling of fluid flows

Mohebujjaman

Rebholz

Iliescu

2018

Numerical Methods in Fluids

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Data‐driven variational multiscale reduced order modeling of vaginal tissue inflation

Snyder

McGuire

Mou

et al. 2022

Numer Methods Biomed Eng

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The vagina undergoes large finite deformations and has complex geometry and microstructure, resulting in material and geometric nonlinearities, complicated boundary conditions, and nonhomogeneities within finite element (FE) simulations. These nonlinearities pose a significant challenge for numerical solvers, increasing the computational time by several orders of magnitude.Simplifying assumptions can reduce the computational time significantly, but this usually comes at the expense of simulation accuracy. This study proposed the use of reduced order modeling (ROM) techniques to capture experimentally measured displacement fields of rat vaginal tissue during inflation testing in order to attain both the accuracy of higher-fidelity models and the speed of simpler simulations. The proper orthogonal decomposition (POD) method was used to extract the significant information from FE simulations generated by varying the luminal pressure and the parameters that introduce the anisotropy in the selected constitutive model. A new data-driven (DD) variational multiscale (VMS) ROM framework was extended to obtain the displacement fields of rat vaginal tissue under pressure. For comparison purposes, we also investigated the classical Galerkin ROM (G-ROM). In our numerical study, both the G-ROM and the DD-VMS-ROM decreased the FE computational cost by orders of magnitude without a significant decrease in numerical accuracy. Furthermore, the DD-VMS-ROM improved the G-ROM accuracy at a modest computational overhead. Our numerical investigation showed that ROM has the potential to provide efficient and accurate computational tools to describe vaginal deformations, with the ultimate goal of improving maternal health.

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A finite element reduced‐order model based on adaptive mesh refinement and artificial neural networks

Baiges

Codina

Castañar

et al. 2019

Numerical Meth Engineering

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In this work a reduced order model based on adaptive finite element meshes and a correction term obtained by using an artificial neural network (FAN-ROM) is presented. The idea is to run a high-fidelity simulation by using an adaptively refined finite element mesh, and compare the results obtained with those of a coarse mesh finite element model. From this comparison, a correction forcing term can be computed for each training configuration. A model for the correction term is built by using an artificial neural network, and the final reduced order model is obtained by putting together the coarse mesh finite element model, plus the artificial neural network model for the correction forcing term.The methodology is applied to non-linear solid mechanics problems, transient quasi-incompressible flows, and a fluid-structure interaction problem. The results of the numerical examples show that the FAN-ROM is capable of improving the simulation results obtained in coarse finite element meshes at a reduced computational cost.

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Data-Driven Filtered Reduced Order Modeling of Fluid Flows

Cited by 141 publications

References 77 publications

Physically constrained data‐driven correction for reduced‐order modeling of fluid flows

Physically constrained data‐driven correction for reduced‐order modeling of fluid flows

Data‐driven variational multiscale reduced order modeling of vaginal tissue inflation

A finite element reduced‐order model based on adaptive mesh refinement and artificial neural networks

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