Hybrid Neural Network Reduced Order Modelling for Turbulent Flows with Geometric Parameters
Abstract
:1. Introduction
2. Models and Methods
2.1. The Full Order Problem
2.2. Mesh Motion
 As shown in Section 2.1, also elementwisely, all the equations are written in their physical domain;
 A finite volume mesh does not have a standard cell shape, resulting in an almost randomshaped polyhedra collection;
 Mapping the equations to a reference domain may require the use of a nonlinear map, but this choice would lead to a change in the nature of the equations of the problem (see [23]).
 1.
 Select the control points in the boundaries to be moved and shift their position obeying the fixed motion rule selected for the geometry modification according to the parameter dependent displacement law: they can be either all the points in the boundary or just a fraction of their total amount if the dimension of the mesh is big enough (see Figure 2), since the higher the number of control points, the bigger (and more expensive) the resulting RBF linear problem to be solved;
 2.
 Calculate all the parameters for the RBF to ensure the interpolation capability of the scheme$$\begin{array}{c}\hfill \mathit{\delta}\left({\mathit{x}}_{i}^{b}\right)={\overline{\mathit{\delta}}}_{i}^{b}\phantom{\rule{4pt}{0ex}},\\ \hfill \sum _{i=0}^{{N}_{b}}{\omega}_{i}q\left({\mathit{x}}_{i}^{b}\right)=0\phantom{\rule{4pt}{0ex}},\end{array}$$$$\left[\begin{array}{cc}\mathrm{\Phi}& \mathit{P}\\ {\mathit{P}}^{\mathit{T}}& 0\end{array}\right]\left[\begin{array}{c}\mathit{\omega}\\ \mathit{\alpha}\end{array}\right]=\left[\begin{array}{c}{\overline{\mathit{\delta}}}^{b}\\ \mathit{0}\end{array}\right]\phantom{\rule{4pt}{0ex}},$$
 3.
 Evaluate all the remaining points of the grid by applying Equation (4).
 Equation (4) is used to move not only the internal points of the grid but also the points located on the moving boundaries that are not selected as control points: even if their displacement could be calculated exactly, changing their position by rigid translation while all the points of the internal mesh are shifted by the use of the RBF may lead to a corrupted grid;
 Equation (5) requires the resolution of a dense linear problem whose dimension is equal to ${N}_{b}+d+1$. Thus, the number of control points has to be carefully selected. Fortunately, the resolution of Equation (5) only has to be carried out once, storing all the necessary parameters to be used in the following mesh motions;
 By the use of this mesh motion strategy, one ends up with meshes having all the same topology, which is an important feature when different geometries have to be compared.
2.3. The Reduced Order Problem
2.4. The Reduced Order SIMPLE Algorithm
Algorithm 1: The Reduced Order SIMPLE algorithm 
Input: first attempt reduced pressure and velocity coefficients ${\mathit{b}}^{\u2605}$ and ${\mathit{a}}^{\u2605}$; modal basis functions matrices for pressure and velocity $\Theta $ and $\Psi $ Output: reduced pressure and velocity fields ${\overline{p}}_{r}$ and ${\overline{\mathit{u}}}_{r}$

2.5. Neural Network Eddy Viscosity Evaluation
3. Results
3.1. Academic Test Case
 An input layer, whose dimension is equal to the dimension of the reduced velocity, i.e., 35, plus one for the parameter;
 Two hidden layers of dimensions 256 and 64 respectively;
 An output layer of dimension 25 for the reduced eddy viscosity coefficients.
3.2. Ahmed Body
4. Discussion
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
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Zancanaro, M.; Mrosek, M.; Stabile, G.; Othmer, C.; Rozza, G. Hybrid Neural Network Reduced Order Modelling for Turbulent Flows with Geometric Parameters. Fluids 2021, 6, 296. https://doi.org/10.3390/fluids6080296
Zancanaro M, Mrosek M, Stabile G, Othmer C, Rozza G. Hybrid Neural Network Reduced Order Modelling for Turbulent Flows with Geometric Parameters. Fluids. 2021; 6(8):296. https://doi.org/10.3390/fluids6080296
Chicago/Turabian StyleZancanaro, Matteo, Markus Mrosek, Giovanni Stabile, Carsten Othmer, and Gianluigi Rozza. 2021. "Hybrid Neural Network Reduced Order Modelling for Turbulent Flows with Geometric Parameters" Fluids 6, no. 8: 296. https://doi.org/10.3390/fluids6080296