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A hyperreduced reduced basis element method for reduced-order modeling of component-based nonlinear systems

Mehran Ebrahimi, Masayuki Yano

Abstract

We introduce a hyperreduced reduced basis element method for model reduction of parameterized, component-based systems in continuum mechanics governed by nonlinear partial differential equations. In the offline phase, the method constructs, through a component-wise empirical training, a library of archetype components defined by a component-wise reduced basis and hyperreduced quadrature rules with varying hyperreduction fidelities. In the online phase, the method applies an online adaptive scheme informed by the Brezzi-Rappaz-Raviart theorem to select an appropriate hyperreduction fidelity for each component to meet the user-prescribed error tolerance at the system level. The method accommodates the rapid construction of hyperreduced models for large-scale component-based nonlinear systems and enables model reduction of problems with many continuous and topology-varying parameters. The efficacy of the method is demonstrated on a two-dimensional nonlinear thermal fin system that comprises up to 225 components and 68 independent parameters.

A hyperreduced reduced basis element method for reduced-order modeling of component-based nonlinear systems

Abstract

We introduce a hyperreduced reduced basis element method for model reduction of parameterized, component-based systems in continuum mechanics governed by nonlinear partial differential equations. In the offline phase, the method constructs, through a component-wise empirical training, a library of archetype components defined by a component-wise reduced basis and hyperreduced quadrature rules with varying hyperreduction fidelities. In the online phase, the method applies an online adaptive scheme informed by the Brezzi-Rappaz-Raviart theorem to select an appropriate hyperreduction fidelity for each component to meet the user-prescribed error tolerance at the system level. The method accommodates the rapid construction of hyperreduced models for large-scale component-based nonlinear systems and enables model reduction of problems with many continuous and topology-varying parameters. The efficacy of the method is demonstrated on a two-dimensional nonlinear thermal fin system that comprises up to 225 components and 68 independent parameters.
Paper Structure (26 sections, 6 theorems, 55 equations, 7 figures, 6 tables, 3 algorithms)

This paper contains 26 sections, 6 theorems, 55 equations, 7 figures, 6 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Parameterized nonlinear PDE model problem
  3. Exact problem formulation
  4. Truth problem formulation
  5. Bubble--port decomposition of functions
  6. Truth problem statement
  7. Hyperreduced reduced basis element method
  8. RB problem formulation
  9. HRBE problem formulation
  10. Component-wise RB and RQ training
  11. Generation of archetype-component training solutions
  12. Component-wise RB construction
  13. Component-wise hyperreduction: BRR theory
  14. Component-wise hyperreduction: formulation
  15. Offline and online computational procedure
  16. ...and 11 more sections

Key Result

Lemma 3

Given an $N$-dimensional Euclidean space $\mathbb{R}^N$, we introduce a $C^1$ mapping $G: \mathbb{R}^N \rightarrow \mathbb{R}^N$, $\mathbf{v} \in \mathbb{R}^N$ such that the Jacobian $DG(\mathbf{v}) \in \mathbb{R}^{N \times N}$ is nonsingular, and constants $\varepsilon$, $\delta$, and $L(\alpha)$ s where $\bar{B}(\mathbf{v}, \alpha) \equiv \{ \mathbf{z}: \left\| \mathbf{z} - \mathbf{v} \right\|_2

Figures (7)

  • Figure 1: (a) Top: an archetype component with two local ports and $\widehat{\mathcal{P}}_1 = \{1,2 \}$, Bottom: an archetype component with three local ports and $\widehat{\mathcal{P}}_2 = \{1,2,3 \}$; (b) A system with $N_\mathrm{comp} = 3$ instantiated components and $N_\mathrm{port} = 5$ global ports. In this system, $M(1) = \widehat{1}$, $M(2) = \widehat{2}$, $M(3) = \widehat{1}$, and $\mathcal{P} = \{1,\cdots,5 \}$. Also, $\Omega_1 = \mathcal{G}_1(\widehat{\Omega}_{M(1)};\mu_1)$, $\Omega_2 = \mathcal{G}_2(\widehat{\Omega}_{M(2)};\mu_2)$, $\Omega_3 = \mathcal{G}_3(\widehat{\Omega}_{M(3)};\mu_3)$, $\Gamma_1 = \mathcal{G}_1(\widehat{\gamma}_{M(1), {1}};\mu_1)$, $\Gamma_2 = \mathcal{G}_1(\widehat{\gamma}_{M(1), {2}};\mu_1) = \mathcal{G}_2(\widehat{\gamma}_{M(2), {2}};\mu_2)$, $\Gamma_3 = \mathcal{G}_2(\widehat{\gamma}_{M(2), {3}};\mu_2)$, $\Gamma_4 = \mathcal{G}_2(\widehat{\gamma}_{M(2), {1}};\mu_2) = \mathcal{G}_3(\widehat{\gamma}_{M(3), {1}};\mu_3)$, and $\Gamma_5 = \mathcal{G}_3(\widehat{\gamma}_{M(3), {2}};\mu_3)$.
  • Figure 2: Archetype components in their reference domains. From left to right: rod, bracket, tee and cross. Local ports are shown by red dashed lines.
  • Figure 3: Decay of POD eigenvalues in the RB construction for the bubble space of different archetype components.
  • Figure 4: RQ points of the archetype components for $\delta_\widehat{c} = 10^{2}$ and $\delta_\widehat{c} = 1$.
  • Figure 5: A $3 \times 3$ fin system. In (a), red stars mark the components with a volumetric source term.
  • ...and 2 more figures

Theorems & Definitions (18)

  • Remark 1
  • Remark 2
  • Lemma 3: Brezzi–Rappaz–Raviart theorem
  • proof
  • Corollary 4: Effectivity bound
  • proof
  • Remark 5
  • Proposition 6
  • proof
  • Corollary 7: Absolute error bound
  • ...and 8 more