What is the minimum set of design points to guarantee sufficient information to obtain a satisfactory RSM. Use parameters from DesignModeler (DM) and ANSYS-CFX Pre to explore a wide range of scenarios, based on limited solution runs. Once the required solutions are met, the Response Surface is created (adapted) to find solutions under unvalued conditions.

## Design of Experiments (DoE)

A sample is a scientifically constructed group of ¨individuals¨ that has the same characteristics as the population. What is the minimum set of design points to provide enough information to obtain a satisfactory system surrogate or RSM. LHS is a generalization of Latin square sampling extended to any number of dimensions.

Latin square is a square grid with sampling positions so that there is only one sample in each row and each column. Ana Paula Curty Cuco, 2009 ESSS, South America ANSYS Users Conference, November 2009, Florianópolis, Brazil and material from Curso ModeFrontier, ESTECO, 2009. Ana Paula Curty Cuco, 2009 ESSS, South America ANSYS Users Conference, November 2009, Florianópolis , Brazil y material from Curso ModeFrontier, ESTECO, 2009.

Before building a definitive DoE table for RSM and/or optimization purposes, we may find that our problem has too many input parameters.

## Parameters Correlation to support DoE

If Autostop is enabled, simulations (DPs) stop when levels of mean and standard deviation error reach the specified level or maximum number of samples is reached. An exhaustive investigation of DoE (Pearson, Spearman, etc.) speeds up the optimization process by reducing the number of variables in the parametric analysis. Once the first variable is ranked from smallest to largest (rank), then the ranking is set to the second variable.

Pearson/Spearman correlation coefficients between X and Y for unbounded ranges and when the range of X is restricted to (0,1). Linear) Correlation matrix in ANSYS-DXTM. Source: http://ocw.usu.edu/Civil_and_Environmental_Engineering/Uncertainty_in_Engineering_Analysis/Regression_DataFitting_Part2.pdf. Can be based on linear or quadratic Coefficient of Determination R2 of the full model for a given output parameter vs.

## Response Surface Methods (RSM)

Response Surface Methodologies (RSM), with special attention to ANSYS-DXTM. RSM is typically an empirical relationship between a variable y and a set of independent variables X1, X2, etc.). Typically used in engineering to build approximate surrogates of higher order analytical tools (e.g. FEA, CFD, etc.). Predictions within the space design are called interpolation, while predictions outside of it are called extrapolations and require caution from the user.

Because of its low cost and simplicity, you should always try it first. It assumes a quadratic relationship between the output and the smallest number of inputs given on the selected hyperplanes, assuming that such DPs correctly represent the output. Once this reduced set of DPs is selected, a quadratic training function is applied to fit the RS.

Once an RSM has been run, clicking on an Output Parameter gives the Goodness-to-fit option to the RSM, based on the current DPs, but we can also create Verification Points (VPs) to test suitability. For example, the root mean square error is not assessed graphically because it is not bounded. VPs are located as far away from DPs as possible by the algorithm, but are not used to build the RSM.

Once the RSM is generated, the VPs are run and compared to the RSM predictions to verify suitability.

Goodness of Fit results

## Plots of RSM

Multi-Objective Optimization

### Multi-Objective Optimization

When there is more than one objective function, but they do not conflict, it means that maximizing one function leads to maximizing the other, and vice versa. The constraints are quantities or limits that are mandatory for the project, for example limits or limitations related to functionality, standards, etc. Constraints can be dimensional (input variable), but they can also be related to output variables (e.g. drag, lift, etc. .). .).

In this case, the GDO will generate its own DPs using the known DoE and RSM techniques. Emulates the evolutionary principles of living systems and obeys Darwin's idea of ¨survival of the fittest¨. Genetic algorithms belong to the more general family of evolutionary algorithms (EA), which generate solutions using a metaheuristic model (based on learning from experience, rule of thumb, trial-and-error, etc.).

These methods have the ambition to solve optimization problems for which we do not know a polynomial algorithm. Based on a hybrid variant of the Non-dominated Sorted Genetic Algorithm-II (NSGA-II), used for continuous variables. NSGA-II is a multiple objective algorithm based on continuous variables, while original MOGA is for discrete spaces.

PROS: high robustness (in terms of finding global critical points) and good at handling multi-objective problems. Can only handle an output parameter target; however, other output limits can be handled via constraints.

Six Sigma Analysis (SSA) and Robust Design

### Six Sigma Analysis (SSA) and Robust Design

The robustness of a solution is defined as the response quality that should be insensitive to variation in input parameters. The user must specify Xmin, Xmax and applies to cases with similar probability for all possible values of the random variable. Applies to the scattering of truly random variables, when lower and higher limits are set by quality control.

This case mainly occurs in random variables that follow the normal distribution after being subjected to a linear operation (eg, subtraction of a geometric quantity). The user must provide the characteristic Weibull parameter Xchr, the Weibull exponent “m”, and the minimum value Xmin (m=2 gives the Rayleigh distribution). This assumption is made on the basis that the input parameters are independent variables by definition (note from LRRS) .