Reduced Order Models for CFD Problems in Finite Volume Setting: The Laminar Case
Reduced Order Models (ROMs) have been developed in order to offer a reduction of the computational burden associated with numerical simulations performed with classical discretisation methods such as the finite element, the finite difference and the finite volume.
In this article we focus on ROMs for problems in Computational Fluid Dynamics (CFD) discretised with the Finite Volume Method (FVM). ROMs have been used extensively in the context of Parametric Partial Differential Equations (PPDEs). In several occasions, one may have to compute an output of interest for several different input parameters. Solving the PDEs system for each value of these parameters could be prohibitive. Therefore, ROMs have been proposed for the goal of computing the output of interest in efficient and accurate manner.
The main assumption of ROMs is that the dynamics of system under study (in our case the Navier — Stokes equations) are governed by a reduced number of dominant modes. These modes embed a large part of the system information. The last assumption translates to another assumption on the solution fields (in our case the velocity and the pressure fields) which permit to write the fields as a finite sum of the modes multiplied by scalar coefficients (essentially the solution fields are expressed as linear combination of their corresponding dominant modes).
At this point the question that has to be raised is how to compute the modes and the coefficients in a way that assures an accurate reconstruction of the fields. Here comes an important notion in reduced order modelling which is the decoupling of two stages named the offline and the online stages.
The offline stage involves computations which are deemed expensive in terms of computational cost. These computations depend on the dimension of the full order model or in other words the number of cells in the finite volume mesh. However, the offline stage is done only once in the life of the ROM. As for the online stage, it involves fast computations which are independent of the dimension of the original problem.
The offline stage consists in the computation of the modes which are called reduced modes/basis. It also consists of the computation of all other terms which depend on the modes. The offline stage is summarized in the flow diagram shown below.
In the diagram above the Proper Orthogonal Decomposition (POD) method is used for the generation of the reduced modes. The POD reduced space is constructed by solving a minimisation problem. The ROMs considered in this article are based on Galerkin projection. These ROMs are termed often as POD-Galerkin ROMs.
Now we proceed to describe the online stage of ROMs for the Navier — Stokes equations. We present here three different ROMs which are based on different reduction assumptions for the pressure field. The first one is based on the uniform approach in which we assume that pressure shares the same coefficients of the velocity. On the other hand, the second and the third ROMs assumes a different set of coefficients. The second ROM is based on the use of the Pressure Poisson Equation (PPE) at the reduced level, whereas the third ROM utilises a stabilisation approach known as the supremiser stabilisation method. The three ROMs differ in the number of equations used in the online stage as well as the type of these equations. Below, we report a schematic description of the three ROMs proposed.
The models described in this article have been tested on benchmark cases in CFD. The codes used for running the simulations are available online in ITHACA-FV⁴ which is a library based on the well-known CFD FV-based code OpenFOAM⁵. For greater details on the numerical tests carried out and the performance of the ROMs we refer the reader to the following articles and contributions.
[1] Stabile, G., Hijazi, S., Mola, A., Lorenzi S., and Rozza G. POD-Galerkin reduced order methods for CFD using Finite Volume Discretisation: vortex shedding around a circular cylinder. Communications in Applied and Industrial Mathematics, 8(1):210–236, 2017.
[2] Hijazi, S., Stabile, G., Mola, A., and Rozza, G. In QUIET Selected Contributions, M. D’Elia, M. Gunzburger, and G. Rozza, Eds., vol. 137 of Lecture Notes in Computational Science and Engineering. Springer International Publishing, 2020, ch. Non-intrusive Polynomial Chaos Method Applied to Full-Order and Reduced Problems in Computational Fluid Dynamics: A Comparison and Perspectives, pp. 217–240.
[3] Stabile, G., and Rozza, G. Finite volume POD-Galerkin stabilised reduced order methods for the parametrised incompressible Navier–Stokes equations. Computers & Fluids 173 (Sep 2018), 273–284.
[4] Stabile, G., and Rozza, G. ITHACA-FV — In real Time Highly Advanced Computational Applications for Finite Volumes. http://www.mathlab.sissa.it/ithaca-fv. Accessed: 2018–01–30.
[5] OpenFOAM website. https://openfoam.org/. Accessed: 13–10–2017.
