[Paper Review] Projection-based reduced order models for a cut finite element method in parametrized domains

Efthymios N. Karatzas, Francesco Ballarin+1 more|arXiv (Cornell University)|Jan 12, 2019

Model Reduction and Neural Networks83 references35 citations

TL;DR

This paper proposes a projection-based reduced order model (ROM) coupled with a CutFEM high-fidelity discretization for parametrized elliptic and Stokes problems, enabling accurate solution of PDEs in domains with large geometric variations without remeshing. By combining natural smooth extension with solution transportation to a common background mesh, the method achieves relative errors of 10⁻⁴ for elliptic problems and 10⁻² for Stokes problems, significantly improving basis efficiency over standard extension alone.

ABSTRACT

This work presents a reduced order modelling technique built on a high fidelity embedded mesh finite element method. Such methods, and in particular the CutFEM method, are attractive in the generation of projection-based reduced order models thanks to their capabilities to seamlessly handle large deformations of parametrized domains. The combination of embedded methods and reduced order models allows us to obtain fast evaluation of parametrized problems, avoiding remeshing as well as the reference domain formulation, often used in the reduced order modelling for boundary fitted finite element formulations. The resulting novel methodology is presented on linear elliptic and Stokes problems, together with several test cases to assess its capability. The role of a proper extension and transport of embedded solutions to a common background is analyzed in detail.

Motivation & Objective

• Address the challenge of high computational cost in solving parametrized PDEs with complex or evolving geometries.
• Overcome limitations of traditional reduced order models that rely on reference domain formulations and remeshing for geometric changes.
• Enable efficient solution of parametrized problems with large deformations and topological changes.
• Develop a methodology that avoids the need for mesh regeneration and transformation maps in ROMs.
• Improve the approximation quality of reduced basis spaces for advection-dominated and boundary-imposed problems.

Proposed method

• Use CutFEM as a high-fidelity solver to handle complex, parametrized domains with embedded boundaries.
• Apply proper orthogonal decomposition (POD) to snapshots generated from CutFEM solutions.
• Apply natural smooth extension to map embedded solutions onto a common background mesh.
• Implement a solution transportation procedure on the background mesh to reduce Kolmogorov n-width and accelerate basis convergence.
• Construct the reduced order model via Galerkin projection onto the POD basis enriched with supremizers for inf-sup stability.
• Employ a background mesh for all operations, enabling offline-online decoupling and avoiding domain-specific mappings.

Experimental results

Research questions

• RQ1 Can CutFEM be effectively combined with projection-based ROMs to handle large geometric parametric variations without remeshing?
• RQ2 How does solution transportation on a background mesh improve the approximation quality of the reduced basis compared to natural extension alone?
• RQ3 What is the impact of weakly enforced Dirichlet boundary conditions on ROM accuracy, especially in the presence of boundary layer effects?
• RQ4 To what extent does the combination of extension and transportation reduce the Kolmogorov n-width and improve convergence of the reduced basis?
• RQ5 Can this methodology achieve high accuracy for both elliptic and Stokes problems with minimal basis size?

Key findings

• The combination of natural smooth extension with solution transportation reduces relative errors by nearly an order of magnitude compared to natural extension alone, achieving errors of 10⁻² for velocity and pressure in the Stokes problem with only N = 3 basis functions.
• For the elliptic problem, the method achieves relative errors of the order of 10⁻⁴ even with a small reduced basis size, demonstrating high efficiency.
• The natural smooth extension alone results in relative errors of up to 10⁻¹ for pressure and 10⁻² for velocity, even with N = 50, indicating poor approximation quality without transportation.
• The method reaches a plateau in error reduction beyond a certain basis size, suggesting limitations due to weak enforcement of Dirichlet boundary conditions via Nitsche's method.
• The high-fidelity solution and reduced-order solution show strong agreement in both velocity and pressure fields, with relative errors localized near embedded boundaries, especially for pressure.
• The proposed approach avoids the need for reference domain mapping and remeshing, enabling efficient solution of problems with large parametric deformations.