A Response Surface Based Strategy for Accelerated Compressor Map Computation

Abstract

The aim of the present work is to develop a strategy enabling fast CFD-based computation of compressor maps for aero engines. The introduced process consists of two phases. In the first phase the compressor limits due to surge and choke are identified and approximated by utilizing methods of support vector machine (SVM). These limit lines are refined within an iterative, distance-based approach. Subsequently, in the second phase the three-dimensional shape of the compressor map is approximated by a response surface method (RSM). The process is validated with an application to an industrial 4.5-stage research compressor, where very good agreement between evaluated and approximated values is obtained.

Appendix

In order to solve the optimization problem (20), matrices

$$\begin{aligned} \mathbf {y}=\left[ y_{1}\ldots y_{N}\right] ^{T},\ \mathbf {1}=\left[ 1 \ldots 1 \right] ^{T},\ \mathbf {X}=\left[ y_{1}\mathbf {x}_{1} \ldots y_{N}\mathbf {x}_{N}\right] \end{aligned}$$

(55)

are introduced. The Langrange function (21) then simplifies to

$$\begin{aligned} L= \mathbf {w}^{T}\mathbf {w} - \pmb {\lambda }^{T}\left( \mathbf {1} - \mathbf {X}^{T}\mathbf {w}-b\mathbf {y}\right) \end{aligned}$$

(56)

and the Karush-Kuhn-Tucker conditions, Bestle (1994), read as

$$\begin{aligned}&\frac{\partial L}{\partial \mathbf {w}} = 2\mathbf {w} + \mathbf {X}\pmb {\lambda }= \mathbf {0}, \end{aligned}$$

(57)

$$\begin{aligned}&\frac{\partial L}{\partial b} = \pmb {\lambda }^{T}\mathbf {y} = 0, \end{aligned}$$

(58)

$$\begin{aligned}&\frac{\partial L}{\partial \pmb {\lambda }} = -\left( \mathbf {1} - \mathbf {X}^{T}\mathbf {w}-b\mathbf {y}\right) \ge \mathbf {0}, \end{aligned}$$

(59)

$$\begin{aligned}&\pmb {\lambda } \le \mathbf {0},\ \lambda _{i} \left[ 1- y_{i}\mathbf {x}_{i}^{T}\mathbf {w} - b y_{i} \right] = 0 \quad \forall i. \end{aligned}$$

(60)

Substituting

$$\begin{aligned} \mathbf {w}= -\frac{1}{2}\mathbf {X}\pmb {\lambda } \end{aligned}$$

(61)

from (57) and inserting it into (56) simplifies the Lagrange function to

$$\begin{aligned} L(\pmb {\lambda }) = \frac{1}{4}\pmb {\lambda }^{T}\mathbf {X}^{T}\mathbf {X}\pmb {\lambda }-\pmb {\lambda }^{T}\mathbf {1} - \frac{1}{2}\pmb {\lambda }^{T}\mathbf {X}^{T}\mathbf {X}\pmb {\lambda } \equiv -\frac{1}{4}\pmb {\lambda }^{T}\mathbf {X}^{T}\mathbf {X}\pmb {\lambda }-\pmb {\lambda }^{T}\mathbf {1} \end{aligned}$$

(62)

or Eq. (23), where the term $b\pmb {\lambda }^{T}\mathbf {y}$ vanishes due to Eq. (58). The dual problem summarizes maximization of the Lagrange function w.r.t. $\pmb {\lambda }$, the condition $\pmb {\lambda } \le \mathbf {0}$ from (60) and constraint (58). Substituting the optimal solution (22) in (61) results in the optimal normal vector (24). The optimal offset may be found from any of the active conditions, where $\lambda _{i}<0$ in (60) results in

$$\begin{aligned} b^{*}= \frac{1}{y_{i}} - \mathbf {x}_{i}^{T}\mathbf {w}^{*} \equiv y_{i} - \mathbf {x}_{i}^{T}\mathbf {w}^{*} \end{aligned}$$

(63)

due to $y_{i}=\pm 1$. The mean value of all $N_{a}$ active constraints is then

$$\begin{aligned} b^{*}=\frac{1}{N_{a}} \left( \sum \limits _{i=1}^{N_{a}} y_{i} - \sum \limits _{i=1}^{N_{a}} \mathbf {x}_{i}^{T}\mathbf {w}^{*}\right) . \end{aligned}$$

(64)

Alternatively only $N_{a}=2$ active training points $\mathbf {x}^{+},\ \mathbf {x}^{-}$ on opposite sides of the separation line with $y^{+}=+1$ and $y^{-}=-1$ may be used canceling the first sum in (64) and yielding Eq. (25), Vapnik (1999).