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Proper Orthogonal Decomposition (POD) Galerkin
We begin the review on RBM with the Proper Orthogonal Decomposition (POD) Galerkin method [37, 126, 14, 4, 93, 5]. The POD has been applied to a wide range of applications (turbulence, image processing applications, analysis of signal, in data compression, optimal control, . . . ). We will detail its offline part, its link with the SVD and the online projection stage. We will see that the NIRB two-grid RB can be generated with the POD offline part and still be nonintrusive. There are several forms of POD (classical POD, spectral POD, . . . ), but here we will mainly consider the Snapshots POD algorithm. To sum up, the algorithm is as follows: • In the offline part, the RB is built with several approximations (the snapshots) of the problem 1.1, for several well chosen parameter values. This step consists, first, in forming the snapshots correlation matrix and in retrieving its eigenvalues and its eigenvectors. Then, the RB functions are constructed by linear combinations of the first N eigenvectors with the snapshots, after having sorted the eigenvalues in descending order.
• The online part consists in solving the reduced problem which uses the RB for a new parameter n ∈ G. At the end of the algorithm, a reduced solution for n is created. The full algorithm is detailed in subsection 1.2.1, following its analysis.
POD-Galerkin Projection on the reduced model (Online stage)
During the last step of the offline stage, the RB (Fi h)i=1,…,N ∈ VN h ⊂ Vdiv = {v ∈ Vh : ∇.v = 0} is generated. To predict the velocity u for a new parameter n, a standard method consists in using a Galerkin projection onto this RB.
This stage, which is intrusive, is much faster than HF codes. The assembling reduced order matrices can be computed offline, and therefore only the new problem with these matrices needs to be solved in the online phase. In what follows, the Galerkin-projection for the velocity field does not contain the pressure field. Indeed, in our model problem, we only use the ROM to derived an approximation on the velocity in the reduced space VN h , since here, with the Dirichlet type boundary conditions, the basis functions satisfy both the boundary conditions and the divergence-free constrain of the continuity equation. Let us consider the equation (1.5). Using the equation (1.30) uk h, f (x) = Nå j=1 aj h,k F j (x).
POD Interpolation (PODI)
This method is a non-intrusive version of the Snapshot POD [113, 57]. The offline part to create the basis functions remains the same. Then, a further step is added. It consists in computing the coefficients for all the original snapshots with a projection (defined by (9)). We denote by ahi (μk), i = 1, . . . , N, k = 1, . . . , Ntrain these coefficients. We obtain N pairs (μk, ahi (μk)).
Thanks to a Gaussian process regression, the function that maps the input parameters μk to the coefficients can be reconstructed. This function is then used during the online stage to find the interpolated new coefficients for a new given parameter n ∈ G. Finally, the highdimensional solution is computed by projecting the new coefficients to the original space with the equation (1.15). We present numerical results on the classical method (Figure 1.2) although several enhancements may be added. For instance, a prior sensitivity analysis of the function of interest with respect to the parameters can be done. This preprocessing phase corresponds to the active subspaces property [40, 35].
The two-grid method
This method will be also detailed in the next chapter along with its numerical analysis. It has been developed and analyzed (with Céa’s and Aubin-Nitsche’s lemmas) in the context of FEM in [96]. As explained in the introduction, its name comes from the fact that it uses two meshes. One fine mesh is employed for the construction of the RB, and an coarse mesh is used to approximate the solution with a classical solver, for instance with FEM or with FV schemes. As other RBM, the two-grid method consists in two stages:
• In the first place, the RB functions are prepared in an « offline » stage with the fine mesh, involving a greedy algorithm 2 or a POD procedure 1.
• Then a coarse approximation of the solution for a new parameter value that is of interest to us is computed « online ». This rough approximation is not of sufficient precision but can be calculated with a smaller number of degrees of freedom compared to the fine mesh. This approximation is then L2-projected onto the RB, and other post-processing steps can be added, such as the rectification method 0.2.2 [30], making it possible to notably improve precision.
NIRB two-grid algorithm
This section recalls the main steps of the two-grid method algorithm [96, 30]. We emphasize on the fact that only the velocity fields are approximated in what follows even if the method also works with the pressure. In the Galerkin POD 1.2, the ROM could require a stabilization term [6, 7]. With the two-grid method, if the FEM solutions satisfy the inf-sup condition (1.3) which ensures stability, since it consists in projecting such solutions and not solving a reduced model, there are no need of additional stabilization terms. The variable parameter is still denoted n. Let uh(n) be the solution approximation computed on a fine mesh Th, with a classical method, and respectively uH(n) be the solution approximation computed on the coarse mesh TH.
We briefly recall the NIRB algorithm. Points 1 and 2 are performed in the offline part, and the others are done online.
NIRB two-grid error estimate with FEM solver
In this section, we develop a precise analysis of the two-grid method with a FEM solver. We consider as a model problem the Poisson’s equation (16a)-(16b), presented in the introduction. The bilinear form a(u, v; μ) is symmetric, continuous and coercive, such that Lax-Milgram’ theorem ensures the well-posedness of the solution. The two grids method is recalled in the last chapter, section 1.4.
Here, we use Delaunay triangulation for the meshes, and thus we have a fine triangular mesh Th and a coarse triangular mesh TH, where h and H are respectively the size of the meshes (see (7)). We use a FEM solver to obtain the fine snapshots and the coarse solutions on these meshes.
NIRB with domain singularities
Main idea. In this section, the NIRB method is exploited on singular domains. The twogrid method with FE solver is applied with a new stategy in order to overcome the effects caused by the singularities. There are many studies in the litterature on re-entrant corners and domain singularities [25, 120, 16, 133, 84, 79]. Most of the developed methods are based on mesh adaptive refinement in the vicinity of the singularities to achieve accuracy [120, 1].
While MOR methods have been proposed for various fields in science and engineering, only few approaches have been developed to treat domain singularities [60, 31, 1]. Therefore, the main idea of this section is to employ the NIRB two-grid algorithm with the rectification post-treatment to approximate such problems. We take advantage of the fact that the RB methods are decomposed in two stages. All the techniques in retrieving an accurate approximation are employed in the offline step. Here, we consider refinement methods. Thus, a refinement is only operated during the offline stage. As a result, the computation times linked to such refinement are considerably reduced in the online phase. We focus on domain with re-entrant corners, and apply the NIRB two-grid method on such problems. It allows us to retrieve optimal errors in the energy norm while using a uniform coarse mesh during the online stage. We emphasize that the NIRB advantages are concerned with the size of the coarse mesh employed during the online stage as well as its uniformity. We tested this new approach on the L-shape domain and on the backward-facing step 3.1, and we present several numerical results. Let us summarize the offline/online strategy:
1. During the “offline” stage, the fine mesh is refined in the vicinity of the re-entrant corners, in order to obtain optimal results. This fine mesh is employed to generate the snapshots. Then the RB is created with the greedy algorithm 2 or the Snapshots POD 1.
2. Then, in the “online” phase, the coarse approximation is computed on a uniform coarse mesh. Afterwards, the coarse solution is projected onto the basis space XN h .
This process allows us to retrieve an accurate approximation with less degrees of freedom. For instance, with the backward-facing step 3.1, the recirculation around the step is very well captured.
Table of contents :
0.1 Context and Motivation
0.1.1 Industrial context
0.1.2 Motivation
0.2 Mathematical tools
0.2.1 Which tools for the simulation of parameter-dependent PDEs?
0.2.2 The two-grid method
0.3 Layout of the thesis
0.3.1 Chapter 1: State of the art
0.3.2 Chapter 2: Numerical analysis of the two-grid method
0.3.3 Chapter 3: Development of new non-intrusive tools
0.3.4 Chapter 4: An offshore wind farm application
0.3.5 Some conclusions and perspectives
1 A review on Reduced Basis methods (RB)
1.1 A model problem
1.2 Proper Orthogonal Decomposition (POD) Galerkin
1.2.1 Proper Orthogonal Decomposition: Offline stage
1.2.2 POD-Galerkin Projection on the reduced model (Online stage)
1.3 POD Interpolation (PODI)
1.4 The two-grid method
1.4.1 NIRB two-grid algorithm
1.5 Numerical results
2 A Non-Intrusive Reduced Basis method: the two-grid method
2.1 NIRB two-grid error estimate with FEM solver
2.1.1 NIRB results with FEM on the model problem
2.2 Some complements on the analysis
2.2.1 Estimate on two different fine meshes
2.2.2 Parabolic equations
2.3 NIRB with domain singularities
2.4 The two-grid method with FV solvers
2.4.1 Main steps
2.4.2 NIRB error estimate
2.4.3 Results on other FV schemes
2.4.4 Some details on the implementation and numerical results
3 New NIRB tools
3.1 Description of the model problem
3.2 NIRB constrained version
3.3 A double reduced basis method based on a domain truncation
3.3.1 Full algorithm
3.3.2 Numerical results on the model problem
3.3.3 Conclusion
4 An industrial application: Offshore wind farm simulations
4.1 Context
4.1.1 Offshore wind farms
4.1.2 Code_Saturne and RANS equations
4.2 The two-grid method applied to the wind farm simulations
4.2.1 2D wind turbine results
4.2.2 3D wind turbine results
A Short review on other NIRB methods
A.0.1 Empirical Interpolation Method (EIM)
A.0.2 Proper Generalized Decomposition (PGD)
A.0.3 Hyper-reduction
B On the “a posteriori” study
C Reminders on TPFA and HMM methods
C.0.1 Two-Point Flux Approximation Finite Volumes (TPFA)
C.0.2 HMM convergence
C.1 Reminder on the Bramble-Hilbert’s lemma
D On the minimization problem
E On the wind farm application
E.1 Mass and momentum equations
E.2 RANS equations
E.3 More results on the 3-dimensional case
E.3.1 3 turbines in line
F GMSH mesh generator
Bibliography
