Preserving Lagrangian Structure in Nonlinear Model Reduction with Application to Structural Dynamics

Carlberg, Kevin; Tuminaro, Raymond S.; Boggs, Paul T.

doi:10.1137/140959602

Cited by 113 publications

(97 citation statements)

References 26 publications

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“…We will see some support for this hypothesis in the numerical examples to follow. Second, there is recent research in model reduction that can ensure that a ROM inherit the same conservation properties that were defined on the original model [27][28][29][30][31][32][33]. These structure-preserving ROMs could be used in place of the POD-DEIM-Galerkin approach.…”

Section: Reduced-order Modeling Of the Ldomentioning

confidence: 99%

Model Structural Inference Using Local Dynamic Operators

DeGennaro

Urban

Nadiga

et al. 2019

Int. J. UncertaintyQuantification

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This paper focuses on the problem of quantifying the effects of model-structure uncertainty in the context of timeevolving dynamical systems. This is motivated by multi-model uncertainty in computer physics simulations: developers often make different modeling choices in numerical approximations and process simplifications, leading to different numerical codes that ostensibly represent the same underlying dynamics. We consider model-structure inference as a two-step methodology: the first step is to perform system identification on numerical codes for which it is possible to observe the full state; the second step is structural uncertainty quantification (UQ), in which the goal is to search candidate models "close" to the numerical code surrogates for those that best match a quantity-of-interest (QOI) from some empirical dataset. Specifically, we: (1) define a discrete, local representation of the structure of a partial differential equation, which we refer to as the "local dynamical operator" (LDO); (2) identify model structure non-intrusively from numerical code output; (3) non-intrusively construct a reduced order model (ROM) of the numerical model through POD-DEIM-Galerkin projection; (4) perturb the ROM dynamics to approximate the behavior of alternate model structures; and (5) apply Bayesian inference and energy conservation laws to calibrate a LDO to a given QOI. We demonstrate these techniques using the two-dimensional rotating shallow water (RSW) equations as an example system. operator (LDO), which is simply a functional relationship between spatially-local field values that approximates the discretized governing field dynamics at a spatial point. For example, if the governing equations are hyperbolic, then the LDO is a function that takes field values in a spatially-local neighborhood of a center point and outputs the field value at that center point, one time step forward in time. Note that there is an attractive consequence of our assumptions of locality and spatio-temporal invariance with respect to system identification: if we wish to infer a LDO from numerical/experimental data, access to the full global state vector is not strictly required. We may simply collect data from a subset of spatial points (together with the appropriate surrounding local neighborhoods). This is a notable advantage relative to a system identification technique that would require the full global state vector (e.g., POD).As all of the dynamics are encoded in the LDO, any structural uncertainties are as well. Furthermore, if we can design the LDO to be a weighted sum of different elementary functions of the local field values, then the relevant structural uncertainties manifest themselves as uncertainties in the values of the weights, which are simply parameters. This is a sketch of the process by which we convert structural uncertainties to parametric ones.Having formalized a means to parameterize model structure, all of the machinery of parametric uncertainty quantification (UQ) is available to study the structural uncert...

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Section: Reduced-order Modeling Of the Ldomentioning

confidence: 99%

Model Structural Inference Using Local Dynamic Operators

DeGennaro

Urban

Nadiga

et al. 2019

Int. J. UncertaintyQuantification

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show abstract

“…More recently, physical constraints have started to be used in standard (ie, without ROM) LES closure modeling (see, eg, the works of Duraisamy et al and Wang et al). Finally, physical constraints have also been used in standard ROM (ie, without closure modeling) . The CDDC‐ROM proposed in this paper uses physical constraints to improve the physical accuracy of the ROM closure model (ie, the Correction term in the DDC‐ROM).…”

Section: Introductionmentioning

confidence: 99%

“…Finally, physical constraints have also been used in standard ROM (ie, without closure modeling). [45][46][47][48][49][50][51][52] The CDDC-ROM proposed in this paper uses physical constraints to improve the physical accuracy of the ROM closure model (ie, the Correction term in the DDC-ROM).…”

Section: Introductionmentioning

confidence: 99%

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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“…The proper orthogonal decomposition (POD) is the most commonly used reduced order modeling technique in large-scale numerical simulations of complex systems. The stability of reduced order models over long-time integration and the structure preserving properties have been recently investigated in the context of Lagrangian systems [5,6], and for port-Hamiltonian systems [7,8]. For Hamiltonian and dissipative Hamiltonian systems, symplectic model reduction techniques are constructed in [9,10,11].…”

Section: Introductionmentioning

confidence: 99%

Structure preserving reduced order modeling for gradient systems

Yıldız

Uzunca

Karasözen

2019

Applied Mathematics and Computation

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Minimization of energy in gradient systems leads to formation of oscillatory and Turing patterns in reaction-diffusion systems. These patterns should be accurately computed using fine space and time meshes over long time horizons to reach the spatially inhomogeneous steady state. In this paper, a reduced order model (ROM) is developed which preserves the gradient dissipative structure. The coupled system of reaction-diffusion equations are discretized in space by the symmetric interior penalty discontinuous Galerkin (SIPG) method. The resulting system of ordinary differential equations (ODEs) are integrated in time by the average vector field (AVF) method, which preserves the energy dissipation of the gradient systems. The ROMs are constructed by the proper orthogonal decomposition (POD) with Galerkin projection. The nonlinear reaction terms are computed efficiently by discrete empirical interpolation method (DEIM). Preservation of the discrete energy of the FOMs and ROMs with POD-DEIM ensures the long term stability of the steady state solutions. Numerical simulations are performed for the gradient dissipative systems with two specific equations; real Ginzburg-Landau equation and Swift-Hohenberg equation. Numerical results demonstrate that the POD-DEIM reduced order solutions preserve well the energy dissipation over time and at the steady state.

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Preserving Lagrangian Structure in Nonlinear Model Reduction with Application to Structural Dynamics

Cited by 113 publications

References 26 publications

Model Structural Inference Using Local Dynamic Operators

Model Structural Inference Using Local Dynamic Operators

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

Structure preserving reduced order modeling for gradient systems

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