Parameterized Orthotropic Plate Systems: Numerical Methods and Model Reduction — From High-Fidelity C⁰ Interior-Penalty Discretizations to Reduced-Order Models for Plates on Elastic Foundations

This dissertation develops and evaluates a unified high-fidelity and reduced-order framework for parameterized thin orthotropic Kirchhoff–Love plates. Motivated primarily by monolithic and panelized flooring systems on compliant foundations, the formulations also apply to plates with parameter-dependent material, support, loading, thermal, and dynamic properties. Three connected model classes are considered: static mechanics with unilateral plate–foundation interaction, steady thermomechanical bending with displacement-dependent thermal exchange, and transient dynamics. The midsurface may be partitioned into non-overlapping structural panels, with transverse displacement and normal rotation transmitted weakly across physical seams. The static mechanical model combines aligned orthotropic bending, parameterized loads and rigidities, panel transmission, and a regularized unilateral Winkler law distinguishing compression from uplift. The fourth-order problem is approximated by a symmetric C⁰ interior-penalty Galerkin method on H¹-conforming Lagrange spaces. Numerical stabilization on element facets is kept distinct from the displacement and normal-rotation penalties across physical seams. An active-set Newton method resolves the unilateral contribution using an algebraic residual stopping criterion. The thermomechanical extension couples the plate to steady three-dimensional heat conduction with exposed-face convection, long-wave radiation, absorbed solar flux, and substrate exchange. A centered first moment of the through-thickness temperature increment defines the plate thermal driver θ, measured in K m⁻¹, and the associated orthotropic thermal bending moments. Displacement modifies lower-face heat transfer through a contact–gap exchange law, while the resulting temperature and thermal driver modify mechanical equilibrium. A relaxed partitioned fixed-point iteration alternates the thermal solve, thermal-to-plate reduction, nonlinear plate solve, and displacement relaxation. The transient extension adds transverse translational inertia, initial displacement and velocity, time-dependent loading, optional algebraic Rayleigh damping, and a consistent mass matrix. A common active-degree restriction removes zero-mass entries in panelized systems, and the average-acceleration Newmark method advances the resulting second-order system. Each trajectory is computed with one prescribed effective Winkler coefficient, so the transient study represents a fixed foundation regime with transverse translational inertia only. The full-order models (FOMs) are assessed through complementary numerical verification. The mechanical approximation is compared with a truncated Navier-series reference and an independent three-dimensional solid model, examined through dual-reference spatial refinement, and evaluated against monolithic references for two-, three-, four-, and six-panel layouts. The thermomechanical solver is assessed through an auxiliary analytical comparison, plate-only and coupled refinement, partitioned-iteration histories, and a cost decomposition showing that the three-dimensional thermal stage and thermal-to-plate reduction dominate recurrent coupled-solve time. The transient implementation is checked through analytical and finite element modal comparisons, time-domain references, direct-to-modal agreement, undamped energy conservation, harmonic and localized-load responses, and panel-to-monolithic consistency. Together, these studies provide numerical verification for the tested configurations, discretizations, and parameter values. Because repeated high-fidelity evaluation is expensive, metric-aware Proper Orthogonal Decomposition (POD) provides a common compression framework for reduced-order models (ROMs). For static mechanics, a non-hyper-reduced intrusive POD–Galerkin model is compared with interpolation and regression of POD coordinates. The thermomechanical FOM remains physically coupled, while its ROM assessment uses two independent one-field workflows generated from the same accepted coupled states. The displacement and thermal-driver fields retain separate output spaces, metrics, training-only bases, predictors, units, and error measures while using matched sample assignments within each parameterization. The non-intrusive methods comprise radial basis function and linear POD interpolation (PODI–RBF and PODI–Linear), Gaussian-process and neural-network regression of POD coordinates (POD–GPR and POD–NN), a POD-enhanced deep-learning ROM (POD–DL-ROM), and direct deep-learning ROMs (DL-ROMs). For transient dynamics, displacement is the only reduced output. Complete trajectories are assigned to disjoint training, validation, and test subsets before flattening, and only training trajectories enter POD construction, preprocessing, and model fitting. PODI–RBF, PODI–Linear, POD–GPR, POD–NN, and POD–DL-ROM map physical parameters and query time to displacement coordinates. Time is an independent query coordinate, yielding a time-conditioned, non-causal displacement-field predictor. Across the sampled parameter domains, the displacement and thermal-driver fields exhibit strong low-dimensional structure, although predictive performance depends on the complexity and coverage of the learned parameter-to-output map. In the mechanical study, POD–GPR achieves accuracy comparable to intrusive POD–Galerkin at substantially lower online cost, while the acceleration of the intrusive model remains limited by full-order residual and Jacobian assembly. In the thermomechanical study, POD–GPR gives the lowest mean thermal-driver error across all parameterizations and the lowest displacement error when the response is parameterized by mechanical loading and absorbed solar flux. Direct DL-ROM gives the best displacement predictions when the response is governed by foundation stiffness and boundary temperatures and in the broader multiparameter setting. In transient dynamics, the load-amplitude maps are predicted close to the projection benchmark, whereas stiffness-dependent response is more demanding because changes in modal frequencies and accumulated phase increase the error of the parameter–time coordinate map. Direct POD projection (POD–Proj) is used as a non-predictive compression diagnostic, and transient online timing is reported as a complete-trajectory-to-single-state cost ratio. Overall, the dissertation establishes a coherent hierarchy from parameterized plate modeling and verified C⁰ interior-penalty approximation to intrusive, POD-based non-intrusive, and deep-learning ROMs for many-query prediction.