Time transient Simulations via Finite Element Network Analysis: Theoretical Formulation and Numerical Validation

TL;DR

The paper extends finite element network analysis (FENA) from static to time-domain transient dynamics by introducing super finite network elements (SFNEs) based on bidirectional recurrent neural networks (BRNNs) and a time-domain network concatenation strategy. SFNEs integrate static inputs via In^S and time-dependent inputs via In^D(t), with a BRNN core that learns transient responses and separate NN blocks to initialize forward and backward states, enabling full-field predictions Out(x,t) over a spatial grid. To overcome limited training windows, the authors propose network concatenation in time, optionally powered by model ensembles to stabilize initial state propagation and using a cut-off time t_c to bound error accumulation; this yields predictions far beyond the training horizon while maintaining accuracy. Demonstrations on 1D homogeneous rods and 2D/inhomogeneous beam-like systems show sub-1% relative errors across multiple cases, with speed-ups of several orders of magnitude compared to FE or analytical solutions, highlighting FENA’s potential as a scalable library-based surrogate framework for transient structural analysis.

Abstract

This paper extends the finite element network analysis (FENA) to include a dynamic time-transient formulation. FENA was initially formulated in the context of the linear static analysis of 1D and 2D elastic structures. By introducing the concept of super finite network element, this paper provides the necessary foundation to extend FENA to linear time-transient simulations for both homogeneous and inhomogeneous domains. The concept of neural network concatenation, originally formulated to combine networks representative of different structural components in space, is extended to the time domain. Network concatenation in time enables training neural network models based on data available in a limited time frame and then using the trained networks to simulate the system evolution beyond the initial time window characteristic of the training data set. The proposed methodology is validated by applying FENA to the transient simulation of one-dimensional structural elements (such as rods and beams) and by comparing the results with either analytical or finite element solutions. Results confirm that FENA accurately predicts the dynamic response of the physical system and, while introducing an error on the order of 1% (compared to analytical or computational solutions of the governing differential equations), it is capable of delivering extreme computational efficiency.

Figures (11)

• Figure 1: Schematic showing the architecture of transient super finite network element of FENA. Consider domain $\Omega$ that is subject to the initial condition $I_0$, boundary condition $\Gamma$, and input $F$. $t$ and $x$ represent time and space domains, respectively. Using the general architecture of SFNE, surrogate models of $\Omega$ can be built to predict its response $Out(x,t)$ to $I_0, \Gamma, F$ at $\{t_1, ..., t_{n_t}\}$.
• Figure 2: Schematic illustrations of the homogeneous rods for (a) Case-1 and Case-2: having constant properties ($E$, $A$, and $\rho$) and an axial boundary load $f(t)$ applied at $x=L$. Note that cases-1 and 2 differ in the initial conditions. (b) Case-3: having constant properties ($E$, $A$, and $\rho$), stiffness boundary conditions ($k_1$ and $k_2$), and a uniformly distributed harmonic load $f(x,t)$.
• Figure 3: (a) Training and test loss trend versus training epochs for the network element representing the transient response of rods subject to boundary load and zero initial conditions. (b) Prediction of the network element for the sample problem $rod_1$. Results correspond to the response at $x=0.6~m$ for the excitation frequency $\omega_0 = 294.7~rad/s$, randomly selected from the test dataset. Superscripts $\Box^a$ and $\Box^{net}$ refer to the solutions obtained from the analytical solution and the network element predictions, respectively.
• Figure 4: (a) Trend of network element ensemble training loss versus training epoch. $net_{1,2,3}$ refers to the SFNEs trained for the problem discussed in § \ref{['sec: probelm statement initial BV rod']}. (b) Initial conditions of $rod_2$. (c) The response predicted by the network ensemble compared to the analytical solution of $rod_2$ for $x=0.6~m$. The response is presented in terms of displacement $u$ and velocity $\dot{u}$ profiles; the superscripts $\Box^{net}$ and $\Box^{a}$ indicate the solutions according to the SFNE and to the analytical approach, respectively.
• Figure 5: Network element concatenation. (a) Average model ensemble relative error versus the number of network element models in the ensembles. The test dataset of the first model in the ensemble was used to calculate the average $e_r$ of the ensemble. All 20 models in the ensemble have the same architecture but are trained with training datasets that are sampled differently. The same training hyperparameters are used for all the 20 network element models in the ensemble (see § \ref{['sec: initial value rod training']}). (b) Distribution of the network ensemble test $e_r$ versus the prediction time step. The relative error value remains low in the entire prediction time window. However, it gradually starts increasing when approaching the upper bound of the window. (c) Initial conditions of the sample case $rod_3$. (d) $rod_3$ response calculated by network element concatenation ($\Box^{net}$) compared with the response obtained from the analytical solution ($\Box^{a}$).