Precision

If the increment is small enough we can use the first order Taylor approximation off to say, in Einstein’s notation:

−ri≈ ∂fi

∂wj

(V q(µ), µ)δwj (2.54)

Therefore we can estimate the argument error as:

E=||∂f_i

∂wj

−1

(V q(µ), µ)r_i|| ≈ ||δw|| (2.55) This can be used to address a known problem in greedy methods (mentioned in [46] and [38]) using the residual as error estimators, which is that the residual might not reflect the proper scale of the error. This method requires access to the sensitivities of the residual. The time for the computation of the inverse of the sensitivities might be prohibitive.

two. In [47] the artificial viscosity parameter of SUPG was based on the original article presenting the SUPG method ([49]) while in [48] the authors developed and employed a method that calibrates these parameters in a specific optimal fashion (see article), they also point to [50] for a method that is model independent, requiring no calibration for either VMS or SUPG. It seems that the choice of parameter can affect the long term integration of the ROM: for a SUPG ROM in [47], a stalled airfoil would produce no CD and CL oscillations after the snapshot time interval (meaning the time interval where snapshots were taken) was exceeded. Instead, they converged to a mean value equal to the mean value of the HDM. In [48] snapshots were taken for 1 vortex shedding cycle of a square cylinder, here both SUPG and VMS ROMs remained accurate 40 vortex shedding cycles after the initial conditions, therefore well after the initial snapshot set. For the general case where we may use either RANS, LES or compressible Euler, [48] presents a review of stabilization methods that add dissipation directly to the reduced model in it’s ODE form.

2.4.2 Hybrid Methods

Hybrid methods attempt to marry an HDM and a ROM so as to speed up calculations whilst preserving a precision comparable to that of the full order model. The hybrid ROM that we are about to present achieve this marriage by only reducing the part of the domain that changes the least with the parameters or boundary conditions that will be varied. An example can be found in [51]: to analyze the effects of wind gusts on an airfoil, the response of domain with no airfoil is calculated, and the corresponding snapshots are used to reduce only the far field of the full computation. Near the airfoil a full-order model is used, and the two domains are coupled via a Schwarz method. We will start by presenting this approach first, and will finish by presenting another one which relies on sub-structuring.

Hybrid reduction by domain decomposition has its origins in the report [52] and later the article [48], where the authors sped by up to 20% the DNS calculations of a vortex-shedding square-cylinder, by using a POD-Least Mean Squares ROM during up to 30% of the calculated time interval. But rather than using POD-Least Mean Squares to reduce part of the time, the authors in [17] suggested using it to reduce part of the physical domain. In this case, the authors analyzed a convergent-divergent conduit with the compressible Euler equations. No shocks were present. The divergent end had its geometry and exit pressure varied, while the convergent end remained static. The full domain was cut in two, overlapping domains: one for the convergent part and the other for the divergent one. For the former POD - Least Mean Squares was used and for the latter a full solve was done each time.

To get the complete answer the two domains were coupled via the Schwarz method (see [53] for a brief presentation, or the full book for a complete overview). The authors obtained relative (to the full HDM) errors on the state variables inferior to 0.5 %. A way to build the required snapshot base was presented in [16], in which residual-based greedy sampling was combined with Voronoi tessellation.

Under the European AerGust (Aeroelastic Gust Modeling) program these methods were applied and further developed ([54], [55]) and finally presented in [51]. Here the authors applied it to a multi query analysis of an airfoil hit by gusts, and optimization (drag-reduction) of the front bumper of a car. The

former was done by DNS, the latter by RANS+Spalart-Allmaras. For the airfoil a 65× speedup from Full CFD to Hybrid was observed all the while maintaining excellent accuracy. For the front bumper optimization the Hybrid model took 26% of the time (with snapshot generation included) of the full CFD optimization , and gave a drag reduction of 3.42 %, while the full model gave one of 3.89%. For the optimization part, the authors recommend mesh morphing in lieu of re-meshing due to much faster execution times. Also, no adaptive or greedy sampling was used for the optimization. Developments are still under way, with sampling strategies being tested [56], as part of a doctoral thesis by the same author entitled ”Aerodynamic Shape Optimization through Reduced-Order Modeling in Industrial Problems”.

Sub-structuring POD Structure-wise, in [57] the authors present their previously developed method of POD - sub-structuring or SPOD ([58]) and introduce a new adaptive sub-structuring method (A-SPOD).

Both these methods rely on Craig-Bampton [59] sub-structuring to separate parts of the structure that deform in a primarily elastic manner, and those that do not. For the model reduction in itself the authors applied the POD-Galerkin method to the typical Newton-Raphson iterations encountered in structural mechanics. The authors tested their methods on the horizontal flute forming of sheet metal: in this process a robot presses down on a sheet of metal to bend it permanently along a horizontal line. They simulated this problem for several sheets of varying thickness using full-order models in order to obtain snapshots. Then they applied SPOD and APOD to a sheet whose thickness was different from the rest. In both cases the calculations are done in less than half of the time of the HDM with relative errors for the displacement below 1% when more than 10 modes (ROM orders superior to 10) are used. Both methods have their disadvantages. A-SPOD requires continuous evaluation during solver iterations, therefore going against the black box principle, which states that a MOR method should not need access to the HDM’s code. SPOD requires the user to guess beforehand which zones will remain primarily elastic and which will not. Still, these methods seem very useful for the structures, but not for fluids. Through S-POD, we could use a ROM where the deformation and the strains are the lowest, such as at the root of the wing, and using the HDM at the tip where they are the highest. We have yet to find articles where such methods are used in optimization.

Chapter 3