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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
