DOE :The Concept of Response Level, The concept of a genetic algorithm, Auto Refinement Mechanism

Define the Concept of Response Level

Response surfaces are functions with different natures that can define any of the parameters or output variables in terms of input parameters or variables. In other words, response surfaces can obtain approximate values of the desired output variable or parameter at any point in the analyzed design space without performing a complete solution process.

As mentioned before, several input parameters are defined first. Each input parameter is divided into several sample design points in the definition of design space in the experimental environment. Therefore, a set of design points is created based on input parameters.

Therefore, a set of design points is created based on input parameters. The response surface based on the results of the solution process of each of the combinations of sample design points, using various methods, defines the most appropriate possible function to estimate the value of the desired output parameter based on the value of one or more selected input parameters.

The response surface production mechanism in the ANSYS Workbench software optimization section has six different types:

  • Genetic aggregation
  • (standard response surface – full 2nd order polynomials)
  • Kriging
  • Non-parametric regression
  • Neural network
  • Sparse grid

Figure 1 shows the various methods for generating response surfaces in the ANSYS Workbench software.


Introduction of Genetic Aggregation Model

The genetic aggregation model solves an iterative genetic algorithm to find the best response level for each output variable or parameter. This method selects the best response modes and combines them to produce a mass or density of several response surfaces. Therefore, this model achieves the highest level of response with different settings for each parameter or output variable.

The main goal in this model is to achieve the following three main criteria to achieve the best response level:

  • Accuracy means high compliance with design points in the test environment (DOE points)
  • Reliability means appropriate cross-validation
  • Smoothness is similar to a linear model


The concept of a genetic algorithm

A genetic algorithm is a special technique for the optimization process; it seeks to find the best values ​​of input parameters or variables to achieve the best output parameter. This model of optimization follows an iterative algorithm. The basis of the operation of this algorithm is derived from the subject of biology.

For example, suppose you decide to turn the people of a city into good people. One way is to identify the city’s good people, separate them from the wrong people, and then force them to spread their generation through childbearing. In fact, by doing so, you can change their genetics and continue this process until the city’s entire population is made up of good people.

Therefore, based on the process mentioned, a cycle can be defined; By first considering the initial population of the city (initialization), then defining a function as a measure of the good or bad of each person in the community (fitness assignment), identifying good people from these criteria. We select these people as parents, crossover; now, the born child may have changes in his genetics and distance himself from his parents’ genetics to which there is a genetic change or mutation. Mutation, and finally to the final conditional stage as a measure of genetic measurement for the end or continuation of the cycle (stop criteria); Thus, if we reach the desired standard (true), the process will end, and if the selected criterion is not met (false), we will again go to the stage of measuring the good or bad of the new generation, and again this We continue the cycle.

Figure 2 shows the general structure of a genetic algorithm in general.


It can now be said that the functional principles of the genetic aggregation algorithm for determining the best response surface are based on the general principles of the genetic algorithm mentioned above. Different states of response levels can be assumed to be equivalent to a city’s population. The criterion for measuring the quality or optimality of response levels can be considered equivalent to the criterion for measuring the genes of the people of a town.

Figure 3 shows the genetic density algorithm. This network algorithm consists of seven steps with a conditional condition that we define step by step each step of this algorithm.

Step 1) Initial population: Different response levels are generated, each with its settings.

Step 2) Evaluation: The created response levels should be measured using accuracy criteria. By defining the degree of tolerance, a criterion can be used to measure the contact levels of the contract.

Step 3 (Selection): The quality of each response level is determined using a cross-validation process and a smoothness measure. The best response levels are then selected to be reproduced in the next step.

Step 4) Conditional step (stop?): After completing a complete round and in each iteration step, if the selected response levels can meet one of the two quality requirements stated in the third step or if the number of iterations is limited, Reaches its maximum, the algorithm process ends, and the final results are presented as the optimal state; Otherwise, the selected levels in the third step as the best answers go to the next step and are reproduced.

Step 5 (Reproduction): The best levels selected in the previous step, along with their settings, are selected as a parent (genes) to cross over with each other and jump or change (mutation).

Step 6: Cross-over: If the parent’s selected response levels are similar, the settings for each response level are mixed; whereas if the parent’s response chosen levels are different, a linear combination of both parent response levels is generated.

Step 7 (mutation): Optionally, changes are made to the settings of each response level; in the same way, in genetics, a child born to his parents over time has a genetic mutation.

Step 8 (new population): In this final stage, new response levels are introduced as a new city population generation. These new levels must return to the second step, their quality assessment, and continue this cycle.


The concept of cross-validation

If the number of input data for the model is too large, it increases the complexity of the model and allows calculations to be not easily performed. In such cases, cross-validation is one of how the number of input data can be optimally determined.

In general, there are two methods for evaluating the performance of a model. In the first method, the evaluation is based on the assumptions that the model should apply. In the second method, the evaluation is based on the model’s performance in predicting new values ​​and observations. Not done. In the evaluation of the first type, it is based on the data that have been observed and used in the construction of the model, such as creating a regression model using existing laboratory data of input and output parameters based on the principle of least squares error; However, this estimated model is only possible for the observed data on which the model is based, and its performance cannot be measured for new data not observed at the time of modeling.

While the cross-validation method relies on data that has been obtained and observed, they have not been used at the time of modeling because the goal, in this case, is to use this existing data but to use it. It is not to measure the model’s efficiency to predict new data. Therefore, to fully evaluate the performance of a model and its optimality, we must estimate the model error based on the data left out in the cross-validation discussion. The estimation of this error is called out-of-sample error. It should be noted that the input data or design points used in estimating the output function or response level are called learning points, and the data or design points used in cross-validation to test the function or The estimated level of response are called the checking point.

Thus, cross-validation acts as a means of calculating off-sample error. The number of input data, the lower the error estimation, and the model moves towards validation. Still, if the number of this input data exceeds a certain number, the error estimation grows again. The slowness and degree of validity of the model decrease. Various methods of cross-validation include the leave-one-out method and the K-fold method.

The concept of the leave-one-out method

In this method, only one of the existing n design points is left out of the response level estimation process, and as a result, the response level based on the n-1 residual point is obtained. Then that independent design point is used to change the quality of the response surface so that the amount of response surface error for this single design point is calculated. This is done for each design point in the test environment.

Figure 4 shows an example of a one-item cross-validation method. As shown in the figure, a design point changes the response level at each stage of the validation process.


The concept of the K-fold method

In this method, the design points are divided into k layers with the same volume. This method works the same as the previous one, except several design points are left out. In the experiment design, the number k is considered to be equal to 10, and hence, the number of cross-validation calculations ends in ten iterations.

Figure 5 shows an example of a multi-layer cross-validation method performed for k = 10 layers.


Auto refinement mechanism

Auto refinement can automatically add several design points to the model until the accuracy of the corresponding response surface reaches the user’s desired limits. This mode is used to increase the accuracy of contact surfaces using an iterative process; this means that at each iteration step, one or more design points are automatically added to estimate the contact level. Therefore, the refinement option should be enabled from the table related to the contact levels of the output parameters based on tolerance. Then the tolerance value for each output parameter should be used as the required criterion or limit for the process. Repeat defined.

This value of tolerance represents the maximum value that the output parameter can accept; thus, at each stage of the response level correction process, the maximum value of the desired parameter or output variable is calculated, and its value is compared with other maximum values ​​obtained from other correction steps. As a result, the maximum possible difference between these values ​​is detected. It is also possible to manually define your desired refinement points in the refinement points table.

The refinement section has configuration options. From the output variable combinations section, the time of application of the new modifier point can be determined. The maximum output option adds a modifier point per repetition to amplify the least accurate output. All outputs mean adding one modifier point for each output is non-convergent. The crowding distance separation percentage option is used to define the minimum allowable distance between the points of the created modifier. The number of refinement points option indicates the number of design points created during the formation of the desired response surface. The maximum number of refinement points option is also used to define the maximum number of refinement points generated during the formation of the response surface.

Figure 6 shows the settings section for correction points in the Response Levels section.


Figure 7 shows the iterative process of creating design points. This iterative process continues until it reaches the desired tolerance level. The horizontal axis (x-axis) represents the number of refinement points, and the vertical axis (y-axis) represents the ratio between the maximum predicted error and the tolerance of each output parameter. Convergence occurs when all output parameters are within the convergence threshold.


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