Reduced-order models for parameter dependent geometries based on shape sensitivity analysis

Abstract

Reduced-order models for parameter dependent geometries based on shape sensitivity analysis Hay, A.; Borggaard, J.; Akhtar, I.; Pelletier, D.Contact us / Contactez nous: nparc.cisti@nrc-cnrc.gc.ca. 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.

Model Order Reduction in Fluid Dynamics: Challenges and Perspectives

“…Ad hoc reduced order modelling techniques have recently been proposed for optimal flow control problems [104,108,121], optimal shape design of devices related with fluid flows [6,58,23,88], and the treatment of fluid-structure interaction problems [76,78].…”

“…In [3,4] the ROMs computed at different parameter points were interpolated to obtain a new ROM that was valid also in the intermediate zone between the original parameter points. In [58,59] the parametric sensitivities of the POD modes were computed and added to the snapshot set, which improved the validity of the reduced solutions away from the parametric snapshots. However, in a more involved geometrical parametrization case the ROM failed completely, as it did not converge to the exact solution even when the number of POD modes was increased.…”

On the Sensitivity Analysis of Angle-of-Attack in a Model Reduction Setting

“…Note that the incompressibility constraint (9) is automatically satisfied since each φ j is solenoidal in the decomposition (1), and its associated Lagrange multiplier, the pressure, is eliminated from (8) or (11). Using the orthogonal decomposition in the set of M equations (18) leads to a set of ODEs for the time coefficients q = [q 1 , .…”

“…parameters that define the geometry of the problem.) We demonstrated our shape sensitivity analysis approaches by developing reduced-order models for flows where the parameter described: 1) the orientation of a square cylinder placed in a channel (10,11); 2) the thickness ratio of an elliptic cross-section cylinder (12,13). In the first case, POD mode sensitivities where computed using flow data obtained by the Sensitivity Equation Method and differentiation of the eigenvalue problem.…”

Reduced dimensional Gaussian process emulators of parametrized partial differential equations based on Isomap

“…Physics-based emulation approaches to such problems typically involve a reduced-basis (e.g. via proper orthogonal decomposition) approximation of the original numerical formulation [3][4][5][6][7][8][9][10][11]. Such approaches are attractive since they establish a direct link to the underlying physical principles and provide rigorous estimates of the error.…”