Parallel time-stepping for fluid–structure interactions

Margenberg, Nils; Richter, Thomas

doi:10.1051/mmnp/2021005

Cited by 8 publications

(4 citation statements)

References 44 publications

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“…We believe that this defect stems from a complicated interplay of temporal dissipation and spatial discretization. For a discussion we refer to [36] and also to [37] where the same effect is observed for a parallel time stepping scheme. In the context of the deep neural network multigrid approach it must be investigated if this problem can be resolved by training with high fidelity solutions that come from discretizations that are finer in space and in time, i.e., by working with a finer time step.…”

Section: Numerical Evaluationmentioning

confidence: 99%

Structure preservation for the Deep Neural Network Multigrid Solver

Margenberg

Lessig

Richter

2021

etna

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The simulation of partial differential equations is a central subject of numerical analysis and an indispensable tool in science, engineering, and related fields. Existing approaches, such as finite elements, provide (highly) efficient tools but deep neural network-based techniques emerged in the last few years as an alternative with very promising results. We investigate the combination of both approaches for the approximation of the Navier-Stokes equations and to what extent structural properties such as divergence freedom can and should be respected. Our work is based on DNN-MG, a deep neural network multigrid technique, that we introduced recently and which uses a neural network to represent fine grid fluctuations not resolved by a geometric multigrid finite element solver. Although DNN-MG provides solutions with very good accuracy and is computationally highly efficient, we noticed that the neural network-based corrections substantially violate the divergence freedom of the velocity vector field. In this contribution, we discuss these findings and analyze three approaches to address the problem: a penalty term to encourage divergence freedom of the network output; a penalty term for the corrected velocity field; and a network that learns the stream function and which hence yields divergence-free corrections by construction. Our experimental results show that the third approach based on the stream function outperforms the other two and not only improves the divergence freedom but also the overall fidelity of the simulation.

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Section: Numerical Evaluationmentioning

confidence: 99%

Structure preservation for the Deep Neural Network Multigrid Solver

Margenberg

Lessig

Richter

2021

etna

Self Cite

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show abstract

“…We believe that this defect stems from a complicated interplay of temporal dissipation and spatial discretization. For a discussion we refer to [34] and also to [35] where the same effect is observed for a parallel time stepping scheme. In the context of the deep neural network multigrid approach it must be investigated if this problem can be resolved by training with high fidelity solutions that come from discretizations that are finer in space and in time, i.e.…”

Section: Numerical Evaluationmentioning

confidence: 91%

Structure Preservation for the Deep Neural Network Multigrid Solver

Margenberg,

Lessig,

Richter

2020

Preprint

Self Cite

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The simulation of partial differential equations is a central subject of numerical analysis and an indispensable tool in science, engineering and related fields. Existing approaches, such as finite elements, provide (highly) efficient tools but deep neural network-based techniques emerged in the last few years as an alternative with very promising results. We investigate the combination of both approaches for the approximation of the Navier-Stokes equations and to what extent structural properties such as divergence freedom can and should be respected. Our work is based on DNN-MG, a deep neural network multigrid technique, that we introduced recently and which uses a neural network to represent fine grid fluctuations not resolved by a geometric multigrid finite element solver. Although DNN-MG provides solutions with very good accuracy and is computationally highly efficient, we noticed that the neural network-based corrections substantially violate the divergence freedom of the velocity vector field. In this contribution, we discuss these findings and analyze three approaches to address the problem: a penalty term to encourage divergence freedom of the network output; a penalty term for the corrected velocity field; and a network that learns the stream function, i.e. the scalar potential of the divergence free velocity vector field and which hence yields by construction divergence free corrections. Our experimental results show that the third approach based on the stream function outperforms the other two and not only improves the divergence freedom but in particular also the overall fidelity of the simulation.

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“…This shift in frequency and phase as a function of mesh resolution prevents a direct comparison of results obtained on different levels in a multigrid-type algorithm such as DNN-MG. In [35], a shifted variant of the Crank-Nicolson scheme is discussed to remedy the problem. Since this cannot be used directly for DNN-MG, we instead compare the functional values in shifted intervals [t * , t * + 1], where t * is chosen as the first peak in the lift functional after time t = 9s, identified separately for the coarse mesh solution, DNN-MG and the fine mesh one on level L + 1.…”

Section: Flow Predictionmentioning

confidence: 99%

A neural network multigrid solver for the Navier-Stokes equations

Margenberg,

Hartmann,

Lessig

et al. 2020

Preprint

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We present a deep neural network multigrid solver (DNN-MG) that we develop for the instationary Navier-Stokes equations. DNN-MG improves computational efficiency using a judicious combination of a geometric multigrid solver and a recurrent neural network with memory. The multigrid method is used in DNN-MG to solve on coarse levels while the neural network corrects interpolated solutions on fine ones, thus avoiding the increasingly expensive computations that would have to be performed on there. A reduction in computation time is thereby achieved through DNN-MG's highly compact neural network. The compactness results from its design for local patches and the available coarse multigrid solutions that provides a "guide" for the corrections. A compact neural network with a small number of parameters also reduces training time and data. Furthermore, the network's locality facilitates generalizability and allows one to use DNN-MG trained on one mesh domain also on an entirely different one. We demonstrate the efficacy of DNN-MG for variations of the 2D laminar flow around an obstacle. For these, our method significantly improves the solutions as well as lift and drag functionals while requiring only about half the computation time of a full multigrid solution. We also show that DNN-MG trained for the configuration with one obstacle can be generalized to other time dependent problems that can be solved efficiently using a geometric multigrid method.

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Parallel time-stepping for fluid–structure interactions

Cited by 8 publications

References 44 publications

Structure preservation for the Deep Neural Network Multigrid Solver

Structure preservation for the Deep Neural Network Multigrid Solver

Structure Preservation for the Deep Neural Network Multigrid Solver

A neural network multigrid solver for the Navier-Stokes equations

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