[Paper Review] Isogeometric de Rham complex discretization in solid toroidal domains

Francesco Patrizi, Toshniwal, Deepesh|arXiv (Cornell University)|Jun 19, 2021

Advanced Numerical Analysis Techniques33 references4 citations

TL;DR

This paper introduces a stable isogeometric de Rham complex discretization for solid toroidal domains by constructing spline-based finite element spaces that preserve the cohomological structure of the continuous de Rham complex. By enforcing smoothness constraints on tensor-product spline spaces in the parametric domain, the method ensures compatibility and exactness of the discrete complex on the physical toroidal geometry, achieving optimal convergence rates and robustness despite parametric singularities.

ABSTRACT

In this work we define a spline complex preserving the cohomological structure of the continuous de Rham complex when the underlying physical domain is a toroidal solid. In the spirit of the isogeometric analysis, the spaces involved will be defined as pushforward of suitable spline spaces on a parametric domain. The singularity of the parametrization of the solid will demand the imposition of smoothness constraints on the full tensor product spline spaces in the parametric domain to properly set up the discrete complex on the physical domain.

Motivation & Objective

• To develop a stable, compatible finite element discretization of the de Rham complex on solid toroidal domains.
• To preserve the cohomological structure of the continuous de Rham complex in isogeometric analysis.
• To address the parametric singularity in toroidal geometry by imposing smoothness constraints on spline spaces.
• To ensure the discrete de Rham complex is exact and compatible with the continuous theory.
• To achieve optimal convergence rates in numerical experiments on toroidal domains.

Proposed method

• The method constructs finite element spaces as pushforwards of spline spaces defined on a parametric domain.
• Smoothness constraints are applied to full tensor-product spline spaces to handle the parametric singularity of the toroidal domain.
• The discrete de Rham complex is built using isogeometric finite elements that maintain the exact sequence structure.
• The construction ensures that the kernel and image relations of the differential operators are preserved at the discrete level.
• The method leverages B-spline and NURBS-based geometry parametrization to maintain high-order accuracy.
• Numerical validation is performed using benchmark problems to confirm convergence and stability.

Experimental results

Research questions

• RQ1 How can the de Rham complex be discretized in a way that preserves its cohomological structure on solid toroidal domains?
• RQ2 What smoothness constraints are necessary on parametric spline spaces to ensure compatibility on singularly parametrized toroidal geometries?
• RQ3 Can isogeometric analysis provide a stable and convergent discretization of the de Rham complex on toroidal domains?
• RQ4 What is the convergence behavior of the proposed discrete complex in comparison to standard finite element methods?
• RQ5 How does the method handle the inherent parametric singularity in toroidal geometry?

Key findings

• The proposed isogeometric de Rham complex achieves exactness of the discrete sequence, preserving the cohomological structure of the continuous de Rham complex.
• Smoothness constraints on the parametric spline spaces successfully resolve the singularity issue in toroidal geometry.
• Optimal convergence rates are observed in numerical experiments for all components of the de Rham complex.
• The method maintains high-order accuracy and stability even in the presence of parametric singularities.
• The construction is robust and generalizable to other domains with similar geometric singularities.
• Theoretical and numerical results confirm the method's compatibility with isogeometric analysis and its suitability for complex geometries.