Convergence of spectral methods for nonlinear conservation laws
SIAM Journal on Numerical Analysis
Preserving symmetries in the proper orthogonal decomposition
SIAM Journal on Scientific Computing
Journal of Computational Physics
A PDE sensitivity equation method for optimal aerodynamic design
Journal of Computational Physics
On Efficient Solutions to the Continuous Sensitivity Equation Using Automatic Differentiation
SIAM Journal on Scientific Computing
The complex-step derivative approximation
ACM Transactions on Mathematical Software (TOMS)
A spectral viscosity method for correcting the long-term behavior of POD models
Journal of Computational Physics
Calibrated reduced-order POD-Galerkin system for fluid flow modelling
Journal of Computational Physics
A continuous shape sensitivity equation method for unsteady laminar flows
International Journal of Computational Fluid Dynamics
Journal of Computational Physics
POD-Based reduced-order models with deforming grids
Mathematical and Computer Modelling: An International Journal
An Online Method for Interpolating Linear Parametric Reduced-Order Models
SIAM Journal on Scientific Computing
Enhanced POD projection basis with application to shape optimization of car engine intake port
Structural and Multidisciplinary Optimization
Mode decomposition methods for flows in high-contrast porous media. A global approach
Journal of Computational Physics
Hi-index | 31.45 |
The proper orthogonal decomposition (POD) is widely used to derive low-dimensional models of large and complex systems. One of the main drawback of this method, however, is that it is based on reference data. When they are obtained for one single set of parameter values, the resulting model can reproduce the reference dynamics very accurately but generally lack of robustness away from the reference state. It is therefore crucial to enlarge the validity range of these models beyond the parameter values for which they were derived. This paper presents two strategies based on shape sensitivity analysis to partially address this limitation of the POD for parameters that define the geometry of the problem at hand (design or shape parameters.) We first detail the methodology to compute both the POD modes and their Lagrangian sensitivities with respect to shape parameters. From them, we derive improved reduced-order bases to approximate a class of solutions over a range of parameter values. Secondly, we demonstrate the efficiency and limitations of these approaches on two typical flow problems: (1) the one-dimensional Burgers' equation; (2) the two-dimensional flows past a square cylinder over a range of incidence angles.