The applicability of model order reduction based on proper orthogonal decomposition to problems in dynamic thermoelasticity with multiple subdomains

Abstract

A robust proper orthogonal decomposition technique is applied to develop reduced-order models (ROMs) for time-dependent thermal stress problems that are arbitrarily discretized with multiple sub-domains to provide flexibility and generality in the sense that different spatial methods and different time integration algorithms can be employed in a single analysis. This approach enables large computational savings for model problems with either/both transient thermal and dynamic structural effects by reducing the degrees of freedom with minimal losses to accuracy. The method of snapshots is used to construct a reduced-order basis from a short training simulation of the full-order model (FOM) which selectively preserves only the relevant physical characteristics of the solution. The approach is described in detail for both first- and second-order ordinary differential equations and differential algebraic equations, such as arising from problems with multiple sub-domains, and the solution of the FOM and ROM by the state-of-the-art GSSSS framework of algorithms is described. Numerical examples in thermal transport, quasi-static thermal stresses, and thermally-induced vibrations for single domains and multiple domains via the finite element method (other methods within each sub-domain can also be integrated with FEM but are not discussed here) illustrate the robustness and utility of the proposed methodology.

Keywords:

• Differential algebraic equation
• dynamics
• subdomain
• thermally-induced vibrations
• thermoelasticity
• time integration; reduced-order modeling

Notes

1 Frobenius Norm: The Frobenius norm of a second-order tensor [A] ∈ ℝ^m×n or ℂ^m×n is defined as the square root of the sum of the absolute squares of its elements, $${\Vert \mathrm{A}\Vert }_{\mathrm{F}}:=\sqrt{\sum _{i=1}^m\sum _{j=1}^n{\left|A_{\mathit{ij}}\right|}^2}=\sqrt{\text{tr}\left({\mathbf{A}}^{†}\mathbf{A}\right)}$$