[Paper Review] High-order Solution Transfer between Curved Triangular Meshes

Danny Hermes, Per-Olof Persson|arXiv (Cornell University)|Oct 16, 2018

Dynamics and Control of Mechanical Systems24 references4 citations

TL;DR

This paper presents a conservative high-order solution transfer method between curved triangular meshes using exact integration via Green’s theorem to compute mass matrix products across curved element intersections. The approach achieves optimal $oldsymbol{ ext{O}}(h^{p+1})$ convergence for $p$-th order isoparametric elements, enabling accurate and conservative remapping in high-order Lagrangian and adaptive finite element simulations.

ABSTRACT

The problem of solution transfer between meshes arises frequently in computational physics, e.g. in Lagrangian methods where remeshing occurs. The interpolation process must be conservative, i.e. it must conserve physical properties, such as mass. We extend previous works -- which described the solution transfer process for straight sided unstructured meshes -- by considering high-order isoparametric meshes with curved elements. To facilitate solution transfer, we numerically integrate the product of shape functions via Green's theorem along the boundary of the intersection of two curved elements. We perform a numerical experiment and confirm the expected accuracy by transferring test fields across two families of meshes.

Motivation & Objective

• To develop a conservative solution transfer algorithm for high-order isoparametric curved triangular meshes, which is essential for high-fidelity simulations in fluid dynamics and solid mechanics.
• To address the challenge of non-convex and multi-component intersections arising in curved mesh element overlaps, which complicate traditional remapping methods.
• To ensure global conservation of physical quantities (e.g., mass, energy) during mesh adaptation or remeshing in Lagrangian and ALE methods.
• To extend existing conservative interpolation techniques—previously limited to straight-sided meshes—toward curved, high-order geometries with optimal convergence rates.

Proposed method

• Utilizes Green’s theorem to transform volume integrals of shape function products into boundary integrals along the intersection of curved donor and target elements, enabling exact numerical integration.
• Employs a Bézier triangle representation for curved elements to model curved boundaries and compute intersections via geometric algorithms tailored to quadratic and higher-order elements.
• Applies an expanding front algorithm to systematically identify and process all intersecting element pairs between donor and target meshes, ensuring complete coverage of the domain.
• Performs high-order quadrature on the resulting curved polygonal intersections using precomputed antiderivatives of polynomial products, avoiding quadrature errors.
• Uses the global coordinates basis to represent shape functions, ensuring that the product of donor and target shape functions remains a polynomial, enabling exact coefficient computation and antiderivative evaluation.
• Solves the discrete projection problem via a Galerkin-type $L^2$-minimizing method using the mass matrix on the target mesh, with errors dominated only by linear system solve precision.

Experimental results

Research questions

• RQ1 How can conservative solution transfer be achieved between high-order isoparametric curved triangular meshes where element intersections are non-convex and multi-component?
• RQ2 What numerical integration strategy enables machine-precision conservation of physical quantities during remapping on curved meshes?
• RQ3 Can the expected $ ext{O}(h^{p+1})$ convergence rate be achieved for high-order solution transfer on curved meshes, despite the non-linear geometry?
• RQ4 How does the use of Green’s theorem and antiderivatives of polynomial products improve accuracy and stability compared to standard quadrature on curved domains?

Key findings

• The proposed method achieves $ ext{O}(h^{p+1})$ convergence for solution transfer on $p$-th order isoparametric curved triangular meshes, confirming theoretical expectations in numerical experiments.
• The algorithm maintains global conservation to machine precision by using exact integration via Green’s theorem and exact polynomial antiderivatives.
• For cubic test functions ($oldsymbol{ ext{ζ}}_1$), the method does not achieve exact transfer on curved meshes due to the isoparametric mapping altering the function space, demonstrating the geometric complexity of curved elements.
• The use of the global coordinates basis allows exact computation of polynomial products and their antiderivatives, avoiding quadrature errors and improving robustness.
• The algorithm is inherently local and parallelizable via domain decomposition, as the expanding front method limits data exchange between subdomains.
• The method is robust for both continuous and discontinuous fields, as integration is restricted to the intersection of donor and target elements, preserving local accuracy.