Unleashing the Power of Reduced Order Models in CFD-DEM Simulations

Arash Hajisharifi
SISSA mathLab
Published in
6 min readOct 15, 2023

Understanding the fluid-solid system is crucial to optimize and improve the performance of such systems in the industry. However, accurately simulating these systems is challenging due to the complex interaction between fluid and solid phases. Simulating these systems experimentally is not reasonable due to the operational and instrumentation costs. Therefore, numerical simulations have become a potentially useful tool to support and possibly replace experiments.

Among the various numerical techniques available, the Computational Fluid Dynamics — Discrete Element Method (CFD-DEM) stands out as a powerful tool for exploring such multiphase systems [1,2,3]. However, its computational complexity has historically limited its use in industrial settings due to the extensive computations required to investigate alternative configurations.

In this context, Reduced Order Models (ROMs) [4,5] based on the Proper Orthogonal Decomposition with Interpolation (PODI) [6] could be proposed as a tool able to increase the efficiency of the CFD-DEM simulations in a parametric fashion without a significant loss of accuracy [7,8,9]. To generalize its application, our method combines ROM with Finite Volume (FV) approach.

To explore a variety of possibilities, we also conduct parametric research using the Stokes number. In this context, we considered two variants of the PODI approach: the classic global PODI and the local PODI. In the local PODI, unlike the global PODI, a POD basis is computed for each parameter and the basis functions for new parameter values are found through interpolation [10]. The fluidized bed benchmark problem is used to verify the ROM accuracy and efficiency of CFD-DEM simulations for industrial applications.

Fig. 1: Sketch of the computational domain with the type of boundary conditions employed. The particle distribution refers to t = 1 s and they are colored by the velocity magnitude.

The computational domain and its boundaries are depicted in Fig. 1. Qualitative illustrations of the instantaneous flow field are depicted in Fig. 2 where the first and second rows demonstrate the time evolution of fluid volume fraction and particle position, respectively, for three different time instances: t = 1 s, t = 2.5 s and t = 4 s from left to right. Initially, all the particles are evenly located; when the gas velocity reaches the fluidization velocity, all the particles are suspended by the upward gas. We clearly observe the complex dynamic and interaction of this two-phase flow system.

Fig. 2: Full Order Model (FOM) solution for the fluid volume fraction (a) and particle position (b) at times t = 1 s (first column), t = 2.5 s (second column) and t = 4 s (third column). The particles are colored based on their velocity magnitude

Time Reconstruction

Our ROM technique shows a very good performance both in terms of
efficiency and accuracy in time reconstruction of the Eulerian and Lagrangian flow fields. A qualitative comparison between the Full Order Model (FOM) solution and ROM reconstruction for Eulerian quantity (fluid volume fraction) and Lagrangian quantities (Particle position and velocity) are shown in Fig. 3 and Fig. 4, respectively.

From an efficiency standpoint, ROM offers accelerated simulations; While the FOM simulation takes approximately 1.81E5 s, computing the reduced coefficients only requires 0.54 s. This remarkable efficiency results in a global speed-up of 200'000, revolutionizing the computational speed of the simulations.

In the context of accuracy, we observe in Fig. 3a that even by considering only 50 % of the energy, ROM reconstructs the main patterns of the flow field, proving its ability to capture the main dynamics. However, some structures may remain undetected. Of course, by employing a larger amount of energy, more accurate results could be obtained. For a visual comparison, please refer to the ROM solutions obtained by retaining 90% of the energy (Fig. 3b).

Fig. 3: Time reconstruction of the Eulerian field: fluid volume fraction computed by ROM for two different energy thresholds 50 % and 90 % at times t = 1s (first column), t = 2.5 s (second column) and t = 4s (third column).
Fig. 4: Time reconstruction of the Lagrangian field: Particle position and particle velocity in three directions computed by ROM for two different energy thresholds 50 % and 90 % at times t = 1s (first column), t = 2.5 s (second column) and t = 4s (third column).

The reconstruction of the Lagrangian fields might be challenging due to the need for the reconstruction of six variables simultaneously for each particle, its position and velocity in the x, y, and z directions. The presence of a large number of particles in these simulations can further amplify the complexity of this process.

Despite the challenges, our ROM technique has shown promising results in efficiently and accurately reconstructing Lagrangian fields as shown in Fig. 4. We have been able to reduce the computational cost while achieving notable accuracy in capturing the main patterns of particle trajectories. In particular, the maximum reconstruction error decreases to just 5.2% when retaining 90% of the energy for ROM reconstruction (Fig. 4b).

Parametric Reconstruction

In many industrial applications, including our case, the governing equations are parametric. This means that the solutions depend on a range of parameter values. As a consequence, these simulations can be computationally expensive. With confidence in our ROM approach’s ability to reconstruct the time evolution of Eulerian and Lagrangian fields, we set our goal to explore the multi-parameter space. To this aim, we built a parametric ROM that could account for variations in the Stokes number — a critical governing parameter in the Eulerian-Lagrangian simulations. We also used our novel approach, local PODI, beyond the global PODI method to increase the accuracy of parametrized ROM.

The global PODI is barely able to reconstruct the main patterns of the flow field. The solution is affected by some spurious oscillations. The situation changes using the local PODI which can capture more details and partially damp the unphysical oscillations.

Fig. 5: Time evolution of the L2-norm relative error for the new sample point.

To introduce a more quantitative comparison, the time history of the L-2 norm relative error between FOM and ROM has been computed and reported in Fig. 5 at a new sample point to evaluate the performance of the parametrized ROM. We note that, except for the initial time of the simulation, the local PODI provides an error lower than the global one. The maximum error decreases by about 15 % when one adopts the local PODI. Also, the mean error decreases passing from 17–18% to 12%.

Conclusion

This work presents a non-intrusive data-driven ROM based on a PODI approach for fast and reliable CFD-DEM simulations [10].
We found that our ROM can capture the unsteady flow features with good
accuracy both for Eulerian and Lagrangian variables. We also performed a parametric study with respect to the Stokes number for the Eulerian phase. We demonstrated that our novel local PODI approach outperforms the global one in a parametric framework both in terms of efficiency and accuracy.

References

[1] S. Golshan, R. Sotudeh-Gharebagh, R. Zarghami, N. Mostoufi, and B. Blais. Review and Implementation of CFD-DEM Applied to Chemical Process Systems. Chemical Engineering Science, 2020.

[2] C. T. Crowe, J. D. Schwarzkopf, M. Sommerfeld, and Y. Tsuji. Multiphase Flows with Droplets and Particles.
Number 2. CRC Press, 2011.

[3] A. Hajisharifi, C. Marchioli, and A. Soldati. Interface topology and evolution of particle patterns on deformable
drops in turbulence. Journal of Fluid Mechanics, 933, 2022

[4] E. Bader, M. K ̈archer, M. A. Grepl, and K. Veroy. Certified Reduced Basis Methods for Parametrized Distributed Elliptic Optimal Control Problems with Control Constraints. SIAM Journal on Scientific Computing,
38:A3921–A3946, 2016

[5] P. Benner, S. Grivet-Talocia, A. Quarteroni, G. Rozza, W. Schilders, and L. M. Silveira. System-and Data-Driven Methods and Algorithms. De Gruyter, 2021

[6] T Bui-Thanh, Murali Damodaran, and Karen Willcox. Proper orthogonal decomposition extensions for parametric applications in compressible aerodynamics. In the 21st AIAA applied aerodynamics conference, page 4213, 2003.

[7] S. Li, G. Duan, and M. Sakai. Pod-based identification approach for powder mixing mechanism in Eulerian–Lagrangian simulations. Advanced Powder Technology, 33, 2022.

[8] T. Yuan, P. G. Cizmas, and T. O’Brien. A reduced-order model for a bubbling fluidized bed based on proper orthogonal decomposition. Computers & chemical engineering, 30(2):243–259, 2005

[9] P. G. Cizmas, A. Palacios, T. O’Brien, and M. Syamlal. Proper-orthogonal decomposition of spatio-temporal patterns in fluidized beds. Chemical Engineering Science, 58:4417–4427, 2003.

[10] Hajisharifi, A., Romanò, F., Girfoglio, M., Beccari, A., Bonanni, D. and Rozza, G., 2023. A non-intrusive data-driven reduced order model for parametrized CFD-DEM numerical simulations. Journal of Computational Physics, p.112355.

--

--