PSI - Issue 54

Arvid Trapp et al. / Procedia Structural Integrity 54 (2024) 521–535

530

10

Arvid Trapp / Structural Integrity Procedia 00 (2023) 000–000

(DK)

G xx , 1 ( f ) G xx , 2 ( f ) . . . G xx , R ( f )

(DK) (DK) . . . (DK)

D 1 D 2

D 1 D 2

(DK) . . . (DK)

R r D

( DK ) r

1 k

=

s ( qsDK ) eq

D R

D R

T 1 , T 2 , ..., T R

k

T

G xx , av =

= D

( DK ) av

1 k

(DK)

s ( DK ) eq

(DK)

1 T

D

D av

R r G xx , r ( f ) · T r

av

Fig. 6. ’Train-on-Dirlik’: Calculating equivalent stress amplitude of non-stationary switching processes s ( qsDK ) eq

and stationary Gaussian process

s ( DK ) eq

for synthetic data generation

2 π f Dr √

and ω 0 = 1 − ζ 2 r , which are then applied on the initial broadband PSDs and the NSM to define the input used for generating the training and test data, as depicted in Fig. 8. For training and testing we chose di ff erent sequences for the processing. While in training the data is generated on the response states Y ( t ), for the test data we included lineary systems theory and began with the loads X ( t ), so that this includes e ff ects that would occur in a realistic fatigue assessment scenario, where the frequency discretization may not be optimal for the given modes. The second notable advantage lies in bypassing RFC, as statistical methods such as Dirlik (Dirlik (1985)) can directly estimate fatigue damage according to the Palmgren-Miner elementry accumulation rule. We use its adaptation for quasi-stationary processes s ( qs , DK ) eq = R r s ( DK ) eq 1 k shown in Trapp and Wolfsteiner (2021a). The procedure for generating the output data is depicted in Fig. 6; the output data generation initiates with a quasi-stationary process, characterized by the PSD of each respective stationary Gaussian subprocess and their corresponding time segments. Next, the damage equivalent amplitude is computed using two distinct methods. Firstly, the damage-rate D r for each subprocess is estimated by applying the Dirlik procedure individually to each sub-PSD. These estimated damage-rates are then weighted according to their respective time segments T r , to obtain the total damage D for each subprocess. The damage equivalent amplitude of the resulting non-stationary process is then derived by summing the individual sub damages D r . In the second approach, the initial step is to compute the PSD of the complete quasi-stationary process by summing the individual sub-PSDs, with each being weighted in accordance with its corresponding time segment and then estimating the equivalent stress amplitude from the averaged PSD, following the standard Dirlik procedure under the assumption of stationarity. This section presents essential parameter of the herein presented ANN model, as well as its specific architecture and its training process. The ANN’s architecture found to be optimal is shown in Figure 9. It was determined by varying the number of neurons from two to 64 and the number of hidden layers from one to ten, resulting in a model with 281 learnable parameters distributed across four hidden layers, each containing eight neurons. We chose the mean squared error as metric resp. loss function to evaluate and compare models towards the optimum. For selecting a suiting activation function we were guided by an extensive and up-to-date survey of functions by (Dubey et al. (2022)). Despite the general lack of a solid theoretical framework, this survey describes the most promising options for specific data sets and input-output parameter configurations. The classic hyperbolic tangent and the Rectified Linear Unit (ReLU) function emerged as the most suitable choices. During the training process, both were thoroughly tested, with the ReLU providing superior performance over the hyperbolic tangent function in this particular setup. As common practice in regression tasks, the output neuron has a linear activation function. This means that the output layer operates as a linear combination of its inputs, allowing the model to predict continuous relationships e ff ectively. 3.2. Architecture and training