Response Surface Method:

Response surface method (RSM) is an approximation function based on a set of mathematical techniques (Cornell 1990). RSM is used in design optimization as a tool to build a relationship between independent design variables (input parameters) and the output response (output parameter). The new approximated function is used to predict the objective function based on small number of runs which replaces long computational time of actual system analysis. As a result RSM is an inexpensive design optimization tool which can be applied to reduce the analysis cost of CFD modelling and FEA. In design optimization the RSM aims to minimize or maximize the response using the relation between the output response (y) and design parameters (𝑥1, 𝑥2) (Anderson and Whitcomb 2005).

𝑦 = 𝑓(𝑥1, 𝑥2) + 𝜖 (5.2) Where f(x) represents the elements function and 𝜖 is the approximation error.

In RSM the designer can identify the sensitivity of output response due to design variables variations in design space which is a set of design points created by applying design of experiments (DOE) approach. The response surface can be represented graphically in terms of

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3D plot or contour lines as shown in Figure 5.4 which shows the functional mapping of design variables to output response.

Figure 5.4 Response surface 3D contours (Anderson and Whitcomb 2005).

Due to the fact that the relation between the output response and design parameters is unknown, the response surface method uses an approximation model known as Meta-model or Meta-function to define this relation. In order to guarantee that the approximation function is able to predict the output response with high accuracy, the fitting between this function and actual engineering model needs to check the Goodness of Fit. The Goodness of Fit criteria compares between the predicted values of output function using RSM and DOE as shown in Figure 5.5 to ensure that all response surface results fit the DOE results and close to the diagonal line (Lawrence 2012).

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Figure 5.5 Comparison between conventional and RSM optimization

RSM uses different regression analysis approaches to create the response function. The commonly used RSM approximation Meta-models are described in the following subsections.

5.4.1.1 Standard Response Surface Model:

In this model the RSM uses first order or second order polynomial to predict the output function. The first order model is a screening tool of linear independent parameters function applicable only for flat surface with least squares. It is also known as low order polynomial and mathematically can be expressed as:

𝑦 = 𝐵_𝑜+ ∑𝑘 𝐵_𝑖𝑥_𝑖

𝑖=1 + 𝜖 (5.3)

The second order model is known as higher degree polynomial which is used when there is a curve in the response surface.

𝑦 = 𝐵_𝑜+ ∑𝑘 𝐵_𝑖𝑥_𝑖

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In some engineering designs, the polynomial models are unlikely to be fitted with the actual system especially for nonlinear problems and it is recommended only when there is a smooth variation in the output response like in FEA for stresses and deformation approximations (Van den Braembussche 2008, Lawrence 2012). However, in other design optimization problems, the 2nd order polynomial approximation quality needs to be improved where the Goodness of Fit is significantly bad.

5.4.1.2 Kriging Model:

In this model, the meta-function uses higher order polynomial equations to determine the relation between output function and input parameters. The Kriging model can interpolate all points in DoE to refine all input parameters with internal error estimation. In this model the relative error in all design space can be predicted as:

Predicted Relative Error =100×Predicted relative error for output Parameter_Out

max−Outmin (5.5)

Where the Out_max and Out_min are maximum and minimum output values.

The main disadvantage of Kriging Meta-model is that this model supports only continuous input parameters and it is not applicable for discrete parameters.

5.4.1.3 Sparse Grid Model:

This meta-model works based on grid adaptive algorithm which is used to build a matrix of design points in a repeated process to meet the desired quality. In this model, the refinement for input parameters is performed automatically to identify the significant parameters levels. The application of this model is limited only for continuous type parameters (Lawrence 2012).

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5.4.1.4 Artificial Neural Network (ANN):

Artificial Neural Network (ANN) is a robust mathematical approximation technique that can be used in design optimization to build a non-linear relationship between input and output parameters (Thevenin and Janiga 2008). In turbine design optimization the ANN is used to predict the turbine performance for different new geometries. Using the information in the database, the ANN model can build a relationship between n input parameters of turbine geometry and m output parameters of turbine performance like efficiency, power, etc. (Cravero and Macelloni 2010).

The working principles of ANN are similar to the human brain nervous system. The ANN consists of neurons or nodes which are collections of elementary processing units. These nodes are wired in layers and joined together with special connections called Synapses with different weights (W). Each node has two operations which are the summation and transformation through the activation function. The simple ANN is organized as shown in Figure 5.6 in three layers; input layer, hidden layer, and output layer (Li 1994). The ANN Meta-model is recommended for turbomachinery optimization due to its ability of fast aerodynamic geometry optimization based on GA with less computational time compared with NS solver (Thevenin and Janiga 2008, Hamzaoui, et al. 2015).

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Figure 5.6 Simple artificial neural network (Thevenin and Janiga 2008).