An Efficient PGD Solver for Structural Dynamics Applications

TL;DR

The paper develops an efficient space-time PGD solver for linear elastodynamics by embedding the PGD framework within a Hamiltonian (symplectic) setting and enriching it with Aitken delta-squared acceleration and robust mode orthogonalization. A key innovation is a Ritz-subspace projection that avoids repeated factorization of the spatial operator, yielding a significant reduction in computational cost while preserving accuracy for 3D structural dynamics. The approach produces a symplectic reduced basis and demonstrates favorable convergence, stability, and scalability across large FE discretizations, outperforming conventional PGD in many scenarios. These results indicate that the proposed Hamiltonian-PGD with Ritz projection is a promising route for fast, structure-preserving reduced-order models in elastodynamics and related wave-type problems.

Abstract

We propose in this paper a Proper Generalized Decomposition (PGD) solver for reduced-order modeling of linear elastodynamic problems. It primarily focuses on enhancing the computational efficiency of a previously introduced PGD solver based on the Hamiltonian formalism. The novelty of this work lies in the implementation of a solver that is halfway between Modal Decomposition and the conventional PGD framework, so as to accelerate the fixed-point iteration algorithm. Additional procedures such that Aitken's delta-squared process and mode-orthogonalization are incorporated to ensure convergence and stability of the algorithm. Numerical results regarding the ROM accuracy, time complexity, and scalability are provided to demonstrate the performance of the new solver when applied to dynamic simulation of a three-dimensional structure.

Figures (8)

• Figure 1: Scheme of the test case.
• Figure 2: Evolution in time of the boundary traction $g_{N} \cdot n$ through time.
• Figure 3: Number of iterations for 20 modes without and with Aitken acceleration
• Figure 4: Visualization of the first three temporal modes (normalized) with and without Aiken acceleration, herein denoted $\widetilde{\psi}_{i}^{q}$ and $\psi_{i}^{q}$, respectively.
• Figure 5: (Left) Error between the reference solutions and the SVD or PGD approximations for 244 926 spatial DOF with 50 modes. (Right) Error between the reference solutions and the SVD, PGD or Modal Decomposition approximations for 36 774 spatial DOF with 300 modes. ($y$-axis has log scale)