ABSTRACT
The continuous demand for performance in modern engineering has led to the design of highly complex structures. Analyzing these designs via the Finite Element Method (FEM) requires solving high-dimensional systems with a prohibitive computational cost when multiple solves are needed, such as in structural optimization or non-linear problems. To mitigate this, Reduced Order Models (ROMs) seek approximations in a lower-dimensional space. Traditional global ROMs obtain fast and accurate approximations but lack topological flexibility to handle continuous geometric variations, which impairs an efficient decoupling between offline training and online execution.
To overcome these limitations, this thesis builds upon the Empirical Interscale Finite Element Method (EIFEM), originally introduced for the multiscale analysis of heterogeneous structures. Formulated within the standard FEM paradigm, EIFEM replaces classical shape functions with precomputed operators learned in an offline stage, aligning with data-driven principles. Unlike traditional approaches based on nested hierarchies, EIFEM directly relates fine-scale and coarse-scale behaviors through these operators. Building on this foundation, this work investigates its application to structural beam modeling, solver acceleration, and optimization.
First, the thesis addresses the modeling of complex composite structures using EIFEM, where coarse-scale degrees of freedom coincide with those of a standard beam formulation. This beam Reduced Order Model (bROM) bridges full-field 3D elasticity and reduced 1D models. Kinematic assumptions are no longer prescribed a priori, but learned from offline 3D simulations. The dimensionality reduction inherent to EIFEM yields a data-driven beam element capable of capturing complex behaviors, such as orthotropy, while maintaining full compatibility with standard FEM implementations.
Second, the focus shifts to the acceleration of exact numerical solvers. EIFEM is used as a coarse space preconditioner within Conjugate Gradient (CG) iterative solvers. By exploiting localized static condensation in a reduced space, this strategy drastically decreases the condition number of large-scale systems. This enables fast, full-fidelity solutions on standard desktop architectures without relying on High-Performance Computing (HPC) environments.
Finally, the thesis tackles the computational cost of structural design by extending the EIFEM framework to parametric settings. The main challenge lies in efficiently evaluating interscale operators for varying geometric configurations of the unit cell during optimization. Since directly interpolating all entries of these high-dimensional mappings would be prohibitive, the Discrete Empirical Interpolation Method (DEIM) is used to identify a reduced set of representative entries.
These are interpolated with respect to geometric parameters to reconstruct the full operators. This strategy enables fast, localized evaluations without requiring global matrix reassembly, accelerating the optimization process while preserving the modularity of the framework.In conclusion, this thesis exploits the offline/online paradigm of localized reduced-order modeling using EIFEM, which provides precomputable operators that encode fine-scale behavior. This approach derives data-driven beam models from 3D elasticity, accelerates exact solvers via data-informed coarse spaces, and enables efficient optimization through parametric operator interpolation, thus aligning standard FEM practice with the machine learning paradigm.
PhD Advisors:
CANDIDATE
Mr Raúl Rubio Serrano is a PhD candidate in the Composites and Advanced Materials for Multifuncional Structures (CAMMS) research group, part of the Aeronautical, Marine, Automotive, and Energy Engineering research cluster.