## 3 Results  
Two numerical and one experimental example were discussed. The provided examples verified and demonstrated the effectiveness of the proposed methodology and algorithms. In the numerical examples, an experiment on the physical subsystem was simulated using different nonlinear functions. All the computations were performed on a PC, and the model order reduction measure was the comparison of (offline) simulation time between the FE model and created ANN. MSEs were also presented. A real-time hybrid simulation with a non-simulated experiment was considered as the last, practical example. Numerical Verification of Implementing ANN in Hybrid Simulation on the Example of Beam Supported by Nonlinear Spring  
A system of a beam (square pipe with outer dimension \(B=48\) mm and inner dimension \(b=45\) mm, of length \(2l=1\) m), fixed at point A and supported by a nonlinear spring at point C, presented in Figure 4, was considered as the first numerical example. The beam was loaded perpendicularly, in the middle of its length, by time-dependent force \(F\). The beam was made of aluminum alloy (Young modulus 69 GPa and density 2720 kg/m\({}^{3}\)). The beam was considered as the analytical subsystem and the spring as the physical subsystem in hybrid simulation. Figure 4: Numerical example 1—mechanical system. Figure 3: Algorithm for building, training, and testing artificial neural network (ANN) for hybrid simulation. In order to generate the reference data to train ANN, the beam was modeled using 10 beam finite elements of equal size. The spring was modeled as a one-rod finite element. Therefore, the system got 20 DOFs (10 transitional and 10 rotational). Rayleigh damping model was adapted, where the damping matrix is proportional to stiffness and mass matrices,  
\[\mathbf{C}=\alpha\mathbf{M}+\beta\mathbf{K}, \tag{6}\]  
with coefficients \(\alpha=2\) and \(\beta=0.4\). The time step of the simulation was set to 0.1 ms. The excitation force \(F\) has the following time-course:  
\[\mathbf{F}(t)=-A[\cos(2\pi f\cdot t)-1]. \tag{7}\]  
with parameters: amplitude \(A\) and frequency \(f\). The following assumptions were made: amplitude cannot exceed 250 N and frequency 100 Hz, the nonlinear stiffness of the spring is in the range from 30 to 180 N/mm. Fifty FE simulations were performed for every combination of parameters \(A=25\) N, 50 N, 75 N,..., 250 N and \(k=30\) N/mm, 45 N/mm, 60 N/mm,..., 150 N/mm. Every analysis was performed with swept-frequency \(f\) from 0 to 100 Hz. A total of 449,820 data samples were generated. There was only one nonzero entry in the vectors \(\mathbf{u^{I}}\) and \(\mathbf{r^{P}}\) (corresponding to vertical displacement of the spring). The exciting force vector \(\mathbf{F}\) was represented only by force applied at point B. Therefore, the ANN got five inputs and one output. Two-layer ANN was trained, with a sigmoid activation function in the hidden layer and linear activation function in the output layer. A total of 70% of the reference data were used as training data, 15% as validation data, and 15% as test data. To train the ANN, Levenberg-Marquardt algorithm was utilized [34] with the random initial solution (normalized using the method of [PERSON] and Widrow [35]). When building the ANN according to the algorithm presented in Figure 3, the following parameters were set: \(N_{min}=1\), \(n=20\), \(\text{MSE}_{\text{max}}=10^{-3}\) mm\({}^{2}\) (sum of errors of all four simulations presented below). The ANN got eight neurons in the hidden layer. Its architecture and a plot of training record error values against the number of training epochs are presented in Figure 5. The time of execution of the algorithm presented in Figure 3 was about 2.5 h.  
To verify the obtained ANN, the nonlinear elasticity of the spring was described by the value of normal force generated in the spring \(\mathbf{N_{s}}\) (N) in the function of vertical displacement of point C \(u_{\text{c}}\) (mm).