[Paper Review] Universal Meshes: A new paradigm for computing with nonconforming triangulations

Ramsharan Rangarajan, Adrián J. Lew|arXiv (Cornell University)|Jan 24, 2012

Lattice Boltzmann Simulation StudiesEngineering25 references20 citations

TL;DR

This paper introduces universal meshes—a novel framework for high-order finite element computations on evolving curved domains using a fixed background triangulation. By mapping background mesh triangles to curvilinear elements via closest point projections and vertex relaxation, it enables conforming, high-order discretizations without remeshing, achieving optimal convergence rates and enabling efficient simulation of moving boundary problems with a single universal mesh.

ABSTRACT

We describe a method for discretizing planar C2-regular domains immersed in non-conforming triangulations. The method consists in constructing mappings from triangles in a background mesh to curvilinear ones that conform exactly to the immersed domain. Constructing such a map relies on a novel way of parameterizing the immersed boundary over a collection of nearby edges with its closest point projection. By interpolating the mappings to curvilinear triangles at select points, we recover isoparametric mappings for the immersed domain defined over the background mesh. Indeed, interpolating the constructed mappings just at the vertices of the background mesh yields a fast meshing algorithm that involves only perturbing a few vertices near the boundary. For the discretization of a curved domain to be robust, we have to impose restrictions on the background mesh. Conversely, these restrictions define a family of domains that can be discretized with a given background mesh. We then say that the background mesh is a universal mesh for such a family of domains. The notion of universal meshes is particularly useful in free/moving boundary problems because the same background mesh can serve as the universal mesh for the evolving domain for time intervals that are independent of the time step. Hence it facilitates a framework for finite element calculations over evolving domains while using a fixed background mesh. Furthermore, since the evolving geometry can be approximated with any desired order, numerical solutions can be computed with high-order accuracy. We demonstrate these ideas with various numerical examples.

Motivation & Objective

• To develop a robust, high-order discretization method for planar C²-regular domains immersed in nonconforming triangulations.
• To eliminate the need for remeshing in time-dependent or evolving domain problems by using a single background mesh as a universal mesh.
• To achieve optimal convergence rates in finite element solutions by constructing exactly conforming curvilinear elements over the immersed domain.
• To enable high-order accuracy in problems sensitive to boundary representation, such as plate bending, by accurately approximating curved boundaries.
• To provide a systematic framework for constructing isoparametric mappings and high-order curved finite elements using only perturbations of a background mesh.

Proposed method

• Construct mappings from background mesh triangles to curvilinear elements that exactly conform to the immersed domain using closest point projections of boundary edges.
• Identify triangles with at least one vertex inside the domain and map edges with both vertices outside to the boundary via closest point projection.
• Relax vertices inside the domain and near the boundary away from the boundary to improve element quality and prevent degenerate triangles.
• Interpolate the constructed curvilinear mappings at selected points (e.g., vertices) to recover isoparametric mappings over the background mesh.
• Use a local perturbation map 𝔪ₕ to relax vertices in a narrow neighborhood of the boundary, with the perturbation radius r chosen as a few times the mesh size.
• Ensure the background mesh satisfies geometric conditions (e.g., acute conditioning angle, sufficient resolution) to guarantee existence of homeomorphisms and well-defined projections.

Experimental results

Research questions

• RQ1 Can a fixed background mesh be used to generate high-order, conforming discretizations for a family of evolving, curved domains without remeshing?
• RQ2 How can curvilinear finite elements be constructed on nonconforming triangulations to achieve optimal convergence rates?
• RQ3 What geometric and meshing conditions must a background mesh satisfy to serve as a universal mesh for a class of immersed domains?
• RQ4 How does the accuracy of high-order solutions depend on the fidelity of boundary representation in immersed finite element methods?
• RQ5 Can vertex relaxation near the boundary prevent element quality degradation while preserving geometric conformity?

Key findings

• The method constructs exactly conforming curvilinear finite elements over immersed domains using only vertex and edge perturbations on a nonconforming background mesh, enabling optimal convergence rates.
• By interpolating the curvilinear mappings at mesh vertices, the approach recovers isoparametric mappings that support high-order finite element methods on the same background mesh.
• The same background mesh can serve as a universal mesh for evolving domains over time intervals independent of the time step, eliminating the need for remeshing.
• Vertex relaxation away from the boundary prevents the formation of poorly shaped or degenerate elements, especially when vertices are close to the boundary.
• The approach maintains sparse matrix structures and connectivity patterns due to fixed mesh topology, significantly improving computational efficiency.
• The method is particularly effective for problems where boundary accuracy critically affects solution quality, such as the bending of a simply supported circular plate.