# Finite-Volume High-Fidelity Simulation Combined with Finite-Element-Based Reduced-Order Modeling of Incompressible Flow Problems

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## Abstract

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## 1. Introduction

## 2. Theory

#### 2.1. Governing Equations

#### 2.1.1. FE Discretization

#### 2.1.2. FV Discretization

#### 2.2. Projection of FV Results to FE Space

**Cell centers to cell nodes interpolation**:In the first stage, the velocity and pressure fields that are calculated by OpenFOAM at the center of the control volumes are interpolated onto the control volume vertices using Inverse Distance Weighting (IDW) interpolation before they are stored at VTK-files. Thus, the stored velocity and pressure results on the VTK-files are implicitly assumed to be interpolated as bilinear fields.**Projection to taylor hood FE space**:In the second stage, the velocity and pressure results stored on the VTK-files are projected onto the Taylor–Hood FE space. Herein, we have used ${L}_{2}$-projection for both velocities and pressure.

#### 2.3. Parametric Dependency

#### 2.4. ROM Using POD

#### 2.5. Mixed and Uniform Methods

## 3. Development of the Solver

#### ROM Solver

## 4. Results and Discussion

#### 4.1. High-Fidelity Simulation Setup

#### 4.2. Snapshots Creation

#### 4.3. Spatial Development of Modes/Reduced Basis

#### 4.4. Spectrum and Related Error

#### 4.5. Accuracy of the Mixed and Uniform Methods

#### 4.6. Computational Speedup

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**NACA64: Exaggerated sketch of the domain setup. Solid and dotted lines on the domain boundary represent the Dirichlet boundaries and Neumann boundary condition, respectively. The parameter space consists of a geometrical parameter $-25{}^{\circ}<\phi <25{}^{\circ}$ and a physical parameter $2\mathrm{m}/\mathrm{s}\mathit{g}20\mathrm{m}/\mathrm{s}$. The mapping ${\pi}_{\phi}$ represents the transformation between the physical and reference geometry.

**Figure 2.**NACA64: (

**a**) Each finite element of size $2h$ is divided into four finite volumes of size h, and together with bilinear interpolation of the geometry ensures that the geometrical representation of $\mathsf{\Omega}$ is the same for the FE and FV mesh. (

**b**) Four finite volumes of size h where the dots represent cell-centered velocity and pressure values, the circles represent velocities at the nodes, and squares represents the pressures at the nodes. OpenFOAM produce cell-centered velocities and pressures, whereas the stored velocity and pressure values on the VTK-files are post-processed to cell-vertices (mesh nodes) by means of Inverse Distance Weighting (IDW) and are assumed to vary bi-linearly. (

**c**) A Taylor–Hood FE of size $2h$ which have biquadratic interpolation of velocities (circles) and bilinear interpolation of pressure (squares).

**Figure 3.**NACA64: Schematic of the hexahedral computational mesh used for high-fidelity simulations with $\phi =-\pi \phantom{\rule{-1.111pt}{0ex}}/\phantom{\rule{-0.55542pt}{0ex}}4$. The computational domain $\mathsf{\Omega}$ has a diameter of $D=30\mathrm{m}$ with a chord length of $c=1\mathrm{m}$. ${\mathbb{G}}_{1}$, ${\mathbb{G}}_{2}$, ${\mathbb{G}}_{3}$ has ${N}_{\theta}=\{60,80,100\}$ elements in the angular direction and ${N}_{r}=\{24,36,44\}$ elements in the radial direction. ${\mathbb{G}}_{2}$ is found to produce a mesh-independent solution. A uniform set of nodes are created around the airfoil (${\mathsf{\Gamma}}_{A}$), and similarly around the outer boundary which is circular (${\mathsf{\Gamma}}_{B}$). In between corresponding nodes on the airfoil with Cartesian coordinate ${\mathit{x}}_{A,i}$ and on the outer boundary with Cartesian coordinate ${\mathit{x}}_{B,i}$ (corresponding points have same angular coordinate) the internal nodes are interpolated using the mathematical expression ($({(j/{N}_{r})}^{3}\times {\mathit{x}}_{B,i})+((1-{(j/{N}_{r})}^{3})\times {\mathit{x}}_{A,i}$)), i.e., j are the levels from airfoil (j = 0) to the boundary ($j={N}_{r}$), the exponent ${}^{3}$ represents the strength of the mesh grading and ${N}_{r}$ are the number of elements in the radial direction.

**Figure 4.**NACA64: The parameter space employed in the analysis consists of the physical parameter $2\mathrm{m}/\mathrm{s}\mathit{g}20\mathrm{m}/\mathrm{s}$ and the geometrical parameter $-25{}^{\circ}<\phi <25{}^{\circ}$. The training set for finding the reduced-order basis is a uniform tensor product parameter space consisting of $10\times 11$ values of the parameters $\mathit{g}$ and $\phi $.

**Figure 5.**NACA64: $\theta \left(r\right)$ as a function of r. For the mathematical expression see Equation (22).

**Figure 6.**NACA64: Simulations conducted by (

**a**) steady and the (

**b**) transient solvers under similar operating conditions (conducted using ${\mathbb{G}}_{2}$ at ($Re=20,\mathit{g}=20\mathrm{m}/\mathrm{s},\phi =25{}^{\circ}$)) (

**c**) shows that the two solutions results in differences that are insignificant for our purposes.

**Figure 7.**NACA64: The first six basis functions for the velocity obtained from the FE-based ROM using the ensemble from FE method. The ensemble solution consists of 110 high-fidelity solutions corresponding to the parameter space of $2\mathrm{m}/\mathrm{s}\mathit{g}20\mathrm{m}/\mathrm{s}\phantom{\rule{3.33333pt}{0ex}}\mathrm{and}\phantom{\rule{3.33333pt}{0ex}}-25{}^{\circ}\phi 25{}^{\circ}$. Higher modes represent the global mean behavior of the flow field. Whereas, lower modes show local behavior with increasing vortex fronts and erratic flow distribution with less coherence.

**Figure 8.**NACA64: The first six basis functions for the velocity obtained from the FE-based ROM using the ensemble from FV method. The ensemble solution consists of 110 high-fidelity solutions corresponding to the parameter space of $2\mathrm{m}/\mathrm{s}\mathit{g}20\mathrm{m}/\mathrm{s}\phantom{\rule{3.33333pt}{0ex}}\mathrm{and}\phantom{\rule{3.33333pt}{0ex}}-25{}^{\circ}\phi 25{}^{\circ}$. Higher modes represent the global mean behavior of the flow field. Whereas, lower modes show local behavior with increasing vortex fronts and erratic flow distribution with less coherence.

**Figure 9.**NACA64: The first six basis functions for the pressure obtained from the FE-based ROM using the ensemble from FV method. The ensemble solution consists of 110 high-fidelity solutions corresponding to the parameter space of $2\mathrm{m}/\mathrm{s}\mathit{g}20\mathrm{m}/\mathrm{s}\phantom{\rule{3.33333pt}{0ex}}\mathrm{and}\phantom{\rule{3.33333pt}{0ex}}-25{}^{\circ}\phi 25{}^{\circ}$.

**Figure 10.**NACA64: The first six basis functions for the pressure obtained from the FE-based ROM using the ensemble from FE method. The ensemble solution consists of 110 high-fidelity solutions corresponding to the parameter space of $2\mathrm{m}/\mathrm{s}\mathit{g}20\mathrm{m}/\mathrm{s}\phantom{\rule{3.33333pt}{0ex}}\mathrm{and}\phantom{\rule{3.33333pt}{0ex}}-25{}^{\circ}\phi 25{}^{\circ}$.

**Figure 11.**NACA64: (

**a**) Velocity and pressure energy spectrum plotted for 60 first modes out of 110 high-fidelity snapshots. The marks are given at intervals of 5 modes. (

**b**) Accumulated energy for the 20 first velocity and pressure modes. The marks are given for the first 10 modes and then in intervals of 2 modes.

**Figure 12.**NACA64: Relative ${H}_{1}$ seminorm for velocity and relative ${L}_{2}$ error for pressure. Increasing number of modes to the left. The marks are given for the first 10 modes and then in intervals of 2 modes. The triangle represents the assumed correlation rate between expected error and the relative error.

**Figure 13.**NACA64: Illustration of Error vs degree of freedoms for pressure and velocity for both mixed and uniform methods. The marks are given for the first 10 modes and then in intervals of 2 modes.

**Figure 14.**NACA64: Relative error of aerodynamic lift and drag error computed as a function of DOFs (i.e., number of modes).

**Figure 15.**NACA64: Relative error between POD modes obtained from mixed and uniform methods for velocity and pressure. (Relative ${H}_{1}$ seminorm for velocity and relative ${L}_{2}$ error for pressure.)

**Figure 16.**NACA64: Illustration of the relative FE - FV errors against the parametric space ($\phi ,\mathit{g}$).

Mixed-Velocity | Uniform-Velocity | Mixed-Pressure | Uniform-Pressure | |
---|---|---|---|---|

DoFs | ${\sum}_{\mathit{i}=\mathbf{1}}^{\mathbf{20}}{\mathit{\lambda}}_{\mathit{i}}$ | ${\sum}_{\mathit{i}=\mathbf{1}}^{\mathbf{20}}{\mathit{\lambda}}_{\mathit{i}}$ | ${\sum}_{\mathit{i}=\mathbf{1}}^{\mathbf{20}}{\mathit{\lambda}}_{\mathit{i}}$ | ${\sum}_{\mathit{i}=\mathbf{1}}^{\mathbf{20}}{\mathit{\lambda}}_{\mathit{i}}$ |

1 | 0.5855145451 | 0.5764554385 | 0.7334786512 | 0.7413536071 |

2 | 0.9320673270 | 0.9276187838 | 0.9947750576 | 0.9948088692 |

3 | 0.9819303885 | 0.9801400555 | 0.9993205450 | 0.9993294807 |

4 | 0.9900400584 | 0.9890380980 | 0.9998532791 | 0.9998502524 |

5 | 0.9954221934 | 0.9947546689 | 0.9999682922 | 0.9999676828 |

6 | 0.9975491011 | 0.9971518139 | 0.9999827134 | 0.9999820036 |

7 | 0.9990781402 | 0.9988937420 | 0.9999945064 | 0.9999940851 |

8 | 0.9993870184 | 0.9992443210 | 0.9999970640 | 0.9999968682 |

9 | 0.9996459298 | 0.9995575066 | 0.9999984249 | 0.9999982012 |

10 | 0.9997908626 | 0.9997329171 | 0.9999995168 | 0.9999994521 |

11 | 0.9998805961 | 0.9998491443 | 0.9999996861 | 0.9999996457 |

12 | 0.9999385575 | 0.9999185397 | 0.9999998352 | 0.9999998220 |

13 | 0.9999560220 | 0.9999403728 | 0.9999998957 | 0.9999998900 |

14 | 0.9999725492 | 0.9999616808 | 0.9999999427 | 0.9999999450 |

15 | 0.9999810290 | 0.9999733205 | 0.9999999688 | 0.9999999681 |

16 | 0.9999892611 | 0.9999845707 | 0.9999999772 | 0.9999999772 |

17 | 0.9999926070 | 0.9999891758 | 0.9999999843 | 0.9999999857 |

18 | 0.9999957257 | 0.9999935479 | 0.9999999881 | 0.9999999904 |

19 | 0.9999969535 | 0.9999953122 | 0.9999999918 | 0.9999999940 |

20 | 0.9999979067 | 0.9999968180 | 0.9999999945 | 0.9999999967 |

**Table 2.**NACA64: Illustration for the speed gains for mixed and uniform methods. $Speedup$ is, time taken for high-fidelity divided by the time taken for the reduced solution on average. $Relative\phantom{\rule{3.33333pt}{0ex}}error$ is absolute ${H}_{1}$ seminorm error divided by ${H}_{1}$ seminorm of the velocity of the reference/high-fidelity solution.

♯ DoFs | Speedup | Relative Error (Velocity) | Relative Error (Pressure) | |
---|---|---|---|---|

High-fidelity | 110 | 1 | 0 | 0 |

Uniform method | 5 | 15,370 | $1.0\times {10}^{-1}$ | $1.1\times {10}^{-1}$ |

10 | 4122 | $3.3\times {10}^{-2}$ | $4.9\times {10}^{-2}$ | |

15 | 1972 | $1.0\times {10}^{-2}$ | $1.5\times {10}^{-2}$ | |

20 | 1011 | $3.1\times {10}^{-3}$ | $3.7\times {10}^{-3}$ | |

Mixed method | 5 | 25,981 | $1.1\times {10}^{-1}$ | $1.2\times {10}^{-1}$ |

10 | 6902 | $4.8\times {10}^{-2}$ | $8.8\times {10}^{-2}$ | |

15 | 2936 | $3.5\times {10}^{-2}$ | $7.2\times {10}^{-2}$ | |

20 | 1764 | $3.2\times {10}^{-2}$ | $4.2\times {10}^{-2}$ |

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**MDPI and ACS Style**

Siddiqui, M.S.; Fonn, E.; Kvamsdal, T.; Rasheed, A.
Finite-Volume High-Fidelity Simulation Combined with Finite-Element-Based Reduced-Order Modeling of Incompressible Flow Problems. *Energies* **2019**, *12*, 1271.
https://doi.org/10.3390/en12071271

**AMA Style**

Siddiqui MS, Fonn E, Kvamsdal T, Rasheed A.
Finite-Volume High-Fidelity Simulation Combined with Finite-Element-Based Reduced-Order Modeling of Incompressible Flow Problems. *Energies*. 2019; 12(7):1271.
https://doi.org/10.3390/en12071271

**Chicago/Turabian Style**

Siddiqui, M. Salman, Eivind Fonn, Trond Kvamsdal, and Adil Rasheed.
2019. "Finite-Volume High-Fidelity Simulation Combined with Finite-Element-Based Reduced-Order Modeling of Incompressible Flow Problems" *Energies* 12, no. 7: 1271.
https://doi.org/10.3390/en12071271