A Hybrid Analytics Paradigm Combining Physics-Based Modeling and Data-Driven Modeling to Accelerate Incompressible Flow Solvers
Abstract
:1. Introduction
2. Mathematical Modeling of Incompressible Flows
3. Hybrid Analytics
- Identify computationally stiff and non-stiff parts of the problem,
- Build a data-driven model to solve the stiff part efficiently,
- Establish encoder and decoder approaches for data transfer between the stiff and non-stiff sub-problems.
3.1. Physics-Based Modeling
3.2. Data-Driven Modeling
3.3. Hybrid Analytics ROM (HA-ROM) to Accelerate Incompressible Flow Solvers
- Offline analysis:
- (i)
- Obtain a set of basis functions for representing dynamics of the problem and satisfying the boundary conditions for the stream function.
- (ii)
- Precompute ROM coefficients for the Poisson equation model as
- Online computation:
- (a)
- FOM update: given and fields, solve the advection–diffusion equation (i.e., computationally non-stiff part) and update the vorticity field:
- (b)
- Encoder (ROM ← FOM): transfer data from the full order space to the reduced order space by using the following projection:
- (c)
- ROM: solve the reduced order model of the Poisson equation until reaching a steady-state solution forSince we are interested in a steady-state solution, the Euler scheme can be used:
- (d)
- Decoder (ROM → FOM): transfer data from the reduced order space to the full order space by using the following definition:
- (e)
- Continue to step (a).
4. Results
4.1. Reduced Order Modeling Results for a Canonical Elliptic Problem
4.2. Hybrid Analytics Results for the Taylor–Green Vortex Decaying Problem
5. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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Poisson Problem () | ||
---|---|---|
ROM Model | ||
1.69 × 10 | 6.32 × 10 | |
1.20 × 10 | 4.51 × 10 | |
3.43 × 10 | 1.21 × 10 | |
6.54 × 10 | 3.12 × 10 | |
6.26 × 10 | 3.10 × 10 | |
6.20 × 10 | 3.10 × 10 | |
6.19 × 10 | 3.10 × 10 |
TGV Problem at | ||||
---|---|---|---|---|
Model | CPU Time () | CPU Time () | ||
FOM | 137.9375 s | 2.31 × 10 | 2156.2968 s | 5.77 × 10 |
HA-ROM () | 1.0156 s | 2.42 × 10 | 4.0625 s | 8.39 × 10 |
HA-ROM () | 1.6250 s | 1.05 × 10 | 6.2656 s | 5.70 × 10 |
HA-ROM () | 2.2187 s | 1.20 × 10 | 8.7812 s | 5.38 × 10 |
HA-ROM () | 3.1406 s | 1.65 × 10 | 13.2812 s | 6.27 × 10 |
HA-ROM () | 5.2187 s | 5.08 × 10 | 15.8281 s | 6.01 × 10 |
HA-ROM () | 12.3583 s | 1.64 × 10 | 45.9531 s | 4.68 × 10 |
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Rahman, S.M.; Rasheed, A.; San, O. A Hybrid Analytics Paradigm Combining Physics-Based Modeling and Data-Driven Modeling to Accelerate Incompressible Flow Solvers. Fluids 2018, 3, 50. https://doi.org/10.3390/fluids3030050
Rahman SM, Rasheed A, San O. A Hybrid Analytics Paradigm Combining Physics-Based Modeling and Data-Driven Modeling to Accelerate Incompressible Flow Solvers. Fluids. 2018; 3(3):50. https://doi.org/10.3390/fluids3030050
Chicago/Turabian StyleRahman, Sk. Mashfiqur, Adil Rasheed, and Omer San. 2018. "A Hybrid Analytics Paradigm Combining Physics-Based Modeling and Data-Driven Modeling to Accelerate Incompressible Flow Solvers" Fluids 3, no. 3: 50. https://doi.org/10.3390/fluids3030050