Radiative heat transfer between moving geometries is critical in many high-temperature engineering systems, yet existing simulation approaches are often too computationally expensive. This study addresses this challenge by developing and testing a reduced order modeling framework for a transient, coupled, nonlinear radiative heat transfer problem with spatially varying subsystems.

We implemented a finite element model of two geometries in Python as a test problem, incorporating the nonlinear radiation boundaries, and varying view factors due to the motion between the subsystems. Different model order reduction techniques are applied on the test problem, each facing distinct limitations. The spatially varying nonlinearity caused most of the complexities. This nonlinearity is solely on the boundary, hence we use substructuring to isolate the nonlinearity. The linear internal dynamics can be reduced with well-established reduction techniques, like the Craig-Bampton method. In addition, the nonlinear boundary term is approximated by a feedforward neural network that was trained on simulation data to approximate nonlinear radiation exchange as a function of the temperatures on the radiative boundaries and relative position, enabling us to ignore the view factor computations, which are computationally very expensive.

The proposed reduced order model (ROM), combining Craig-Bampton reduction for internal degrees of freedom with a neural network approximation for interface radiation, achieves significant computational savings compared to the full order model (FOM).
Results show that with decoupling internal and interface dynamics, the Craig-Bampton basis reduces the system matrices sizes while preserving accuracy in the full solution.
On the interface, the neural network provides an efficient approximation for the nonlinear, spatially varying radiation operator, reducing the computational costs from quadraticn, O(NM), scaling (FOM) to a linear, O(N+M), dependence, where N and M are the number of elements on the radiative boundaries of two components.
In time integration, the Newton iterations remain identical in terms of the setup to the full model, but the reduced residual and Jacobian evaluations are significantly faster as the matrix sizes are reduced.
Although both the Craig-Bampton reduction and neural network require additional offline computations (modal analysis and training), these are one-time costs.
The online benefits are significant: the ROM reproduces the full solution very accurately while enabling much faster simulations, making it very useful for repeated evaluations, parametric-, and design studies. ... Read more

This thesis explores the improvement of computational efficiency in simulating two-dimensional conveyor belt systems by applying model order reduction (MOR) techniques. Conveyor belts, crucial for material handling in various industries, are traditionally modeled using finite element methods (FEM), which can be computationally demanding, particularly for long-term simulations with a high number of grid elements. To address this, MOR techniques aim to reduce high-dimensional models into lowerdimensional models.
The study investigates three MOR approaches—modal decomposition, Proper Orthogonal Decomposition (POD), and Dynamic Mode Decomposition (DMD)—to apply on existing software built by VORtech that simulates two-dimensional conveyor belt systems. When considering MOR to reduce the two-dimensional system, challenges arise related to nonlinearities, differential algebraic equations (DAEs), and complex modeling steps. Among the methods tested, DMD proved to be the most effective, offering significant reductions in computational time while maintaining accuracy. POD also demonstrated accuracy but had less impact on speed due to the time-consuming complex modeling steps in the simulation software. These complex modeling steps are not investigated in detail in this thesis and therefore not reducible with the intrusive POD. Because of the non-intrusive nature of DMD, this method was able to incorporate these extra processes in the reduced order model.
The study concludes with recommendations for future research, emphasizing the need for optimization of the code segments that handle the complex modeling steps. In addition, conventional modeling approaches as alternatives to the complex interpolation step could be explored, to enhance the applicability of MOR. Furthermore, DMD with control or parametric DMD could be explored to obtain a reduced order model by interpreting misalignments of rollers as controls or parameters. Finally, a method is proposed to make modal decomposition useful for models, where the solutions depend highly on the external forces. Although this method is not applied to the model considered in this thesis, it would be interesting to explore this modified modal decomposition method on other models that are significantly influenced by the external forces. ... Read more

The computational demands of finite element simulations, particularly in predicting the time-dependent response of high-dimensional non-linear dynamical systems, pose significant challenges. To overcome these challenges, researchers have developed model order reduction (MOR) methods, which aim to reduce computational complexity by utilizing lower-dimensional models. This thesis proposes a MOR technique that simultaneously learns both the projection to, and the reduced dynamics on, a lower-dimensional manifold using autoencoders, a type of neural network. During training, the known linear part of the reduced dynamics is used to aid the optimization process, leading to an effective method of simultaneous projection and linear informed training (SPLIT). SPLIT demonstrates outstanding performance on the test case of a 2D cantilever beam, and is capable of making non-linear forced response predictions, even though being trained on unforced decaying trajectories. Even in scenarios involving highly non-linear behaviour, such as when the beam folds over itself, SPLIT continues to make accurate predictions, while other MOR techniques fail. This work highlights the potential of autoencoders to advance the field of MOR and improve the efficiency and reliability of simulations for complex dynamical systems. ... Read more

This paper examines whether complex high-dimensional data that describes the dynamics of a cantilever beam can be transformed into a less complex system. In particular, the transformation that is examined is the reduction of the dimension. An essential aspect of this study involves the implementation of a linear autoencoder, which is a type of machine learning model that possesses the capability to effectively reduce the dimensionality of input data while adeptly reconstructing the original dataset. The model performs well and is successful in reconstructing complex data via the less complex system. However, the model struggles if the dynamics are made more complex by adding an external force. Although the dynamics seem to be present in the results, the amplitudes differ. ... Read more