A Certified Two-Step Port-Reduced Reduced-Basis Component Method for Wave Equation and Time Domain Elastodynamic PDE

Mohamed Aziz Bhouri

Published 2020-02-25Version 1

We present a certified two-step parameterized Model Order Reduction (pMOR) technique for wave equation and elastodynamic Partial Differential Equations (PDE). pMOR techniques for parameterized time domain PDEs offer opportunities for faster solution estimation. However, due to the curse of dimensionality, basic pMOR techniques fail to provide sufficiently accurate approximation when applied for large geometric domains with multiple localized excitations. Moreover, considering the time domain PDE for the construction of the reduced basis greatly increases the computational cost of the offline stage and treatment of hyperbolic PDEs suffers from pessimistic error bounds. Therefore, within the context of linear hyperbolic time domain PDEs for large domains with localized sources, it is of great interest to develop a pMOR approach that provides relatively low-dimensional spaces and which guarantees sufficiently accurate approximations. Towards that end, we develop a two-step Port-Reduced Reduced-Basis Component approach (PR-RBC) for linear hyperbolic time domain PDEs. First, our approach takes advantage of the domain decomposition technique to develop reduced bases for subdomains, which, when assembled, form the domain of interest. This reduces the effective dimensionality of the parameter spaces and solves the curse of dimensionality issue. Moreover, the time domain solution is the inverse Laplace transform of a frequency domain function. Therefore, we can approximate the time domain solution as a linear combination of the PR-RBC solutions to the frequency domain PDE. Hence, we first apply the PR-RBC method on the elliptic frequency domain PDE. Second, we consider the resulting approximations to form a reduced space that is used for the time solver. We also provide an a posteriori error estimate for the two-step PR-RBC approach based on the time-frequency duality.