PSI - Issue 68

Morteza Khomami Abadi et al. / Procedia Structural Integrity 68 (2025) 1312–1318 Morteza Khomami Abadi et al. / Structural Integrity Procedia 00 (2025) 000–000

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not include other loading and boundary conditions. It should be noted that in addition to the above limitations, the crack depth for a range greater than 0.6-0.7 leads to the change of all equations and requires a new solution, which is a fundamental challenge. As a result, it is necessary to introduce new methods to solve these limitations. 2.2. Group Method of Data Handling, GMDH A set of neurons with various pairs of neurons in each layer connected by a quadratic polynomial can be used to represent the conventional GMDH method, which was developed using the neural network concept by Ivakhnenko. Three components make up the basic structure of the GMDH parts: the input, hidden, and output layers. In order to determine the least squares error between the algorithm's results and the actual output, the hidden layers are generated and the polynomials are extracted. In most cases, the GMDH is used to determine a function based on input and output data. The result vector, Y, can be represented in terms of the input vector, X, using this function: = ( , , " , … , 6 ) , =( , , " , … , 7 ) (4) 8 = ( 8, , 8" , … , 86 ), =1 , 2 , … , (5) The GMDH analysis should minimize the square of the difference between the approximate and real outputs as below: (6) T [( ( 8, , 8" , … , 86 ) − 8 ) " ] ≪ 7 89, Using the relationship between input and output variables, the GMDH method produces a Kolmogorov–Gabor polynomial by Besarati. (7) 8 = ( + T 8 8 + ⋯ + T T 8: 8 : 6 :9, 6 89, 6 89, + T T T 8:; 8 : 6 ;9, 6 :9, 6 89, ; +⋯ The constants ! ( = 0,...,5) used in this equation are obtained in terms of input data containing geometrical parameters, material properties, loading, and boundary conditions, etc. The Kolmogorov-Gabor form's simplified classic case can be presented as 8 = \ 8 , : ] = ( + , 8 + " : + $ 8 " + % : " + < 8 : If the necessary data is available, the GMDH technique can be used to determine complex relationship between output and input variables. 3. Methodology of research In this research, the stress intensity factors of beams and plates are determined without previous limitations based on the following steps: • In the first step, the number of tests required for accurate estimation of stress intensity factors is determined by the Design of Experiment (DOE) technique. • In the second step, a series of numerical modeling by changing the geometric, loading, boundary conditions, etc. is performed in the ABAQUS software to determine the initial values of the stress intensity factors. • In the third step, input variables (geometric, loading, boundary conditions, etc.) and output variables (stress intensity factors extracted from Abaqus software) are analyzed by the GMDH neural network and a relationship is created between the input and output variables. • In the fourth step, the GMDH neural network factorial design , the number of hidden layers, the number of tests, allowable error value, normalization technique, etc. are controlled and the most optimal possible conditions are determined. • In the fifth step, explicit relationships are extracted to determine the stress intensity coefficients. • In the sixth step, the extracted relationships are validated with the references and the efficiency of the relationships is controlled with new variables. In the GMDH algorithm, the DOE technique is used for the optimization of the number of experiments and the collection of the combination of input data. In the DOE technique, the factorial design of two levels combined with (8)