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High-order Solution Transfer between Curved Triangular Meshes

Danny Hermes, Per-Olof Persson

TL;DR

The paper tackles conservative solution transfer between curved 2D meshes by formulating a Galerkin projection where the transfer satisfies $M_T \mathbf{t} = M_{TD} \mathbf{d}$, ensuring conservation when the constant function is representable. It introduces a common refinement of donor/target elements and an expanding-front algorithm to identify all intersecting pairs, while employing Green's theorem to convert curved-polygon integrals into line integrals for exact, high-order evaluation. The key contributions are (i) Bézier-triangle intersection and curved-polygon handling, (ii) an $\mathcal{O}(|\mathcal{M}_D|+|\mathcal{M}_T|)$ intersection search, and (iii) a robust integration strategy over curved regions that yields provable convergence, validated by numerical experiments. By enabling high-order, conservative remapping on curved meshes, the method supports flexible mesh adaptivity and potentially reduces remeshing frequency in ALE/Lagrangian simulations, with clear paths for extending to three dimensions and alternative basis representations.

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.

High-order Solution Transfer between Curved Triangular Meshes

TL;DR

The paper tackles conservative solution transfer between curved 2D meshes by formulating a Galerkin projection where the transfer satisfies , ensuring conservation when the constant function is representable. It introduces a common refinement of donor/target elements and an expanding-front algorithm to identify all intersecting pairs, while employing Green's theorem to convert curved-polygon integrals into line integrals for exact, high-order evaluation. The key contributions are (i) Bézier-triangle intersection and curved-polygon handling, (ii) an intersection search, and (iii) a robust integration strategy over curved regions that yields provable convergence, validated by numerical experiments. By enabling high-order, conservative remapping on curved meshes, the method supports flexible mesh adaptivity and potentially reduces remeshing frequency in ALE/Lagrangian simulations, with clear paths for extending to three dimensions and alternative basis representations.

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.

Paper Structure

This paper contains 21 sections, 1 theorem, 43 equations, 21 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. General Notation
  4. Bézier Curves
  5. Bézier Triangles
  6. Curved Elements
  7. Shape Functions
  8. Curved Polygons
  9. Galerkin Projection
  10. Linear System for Projection
  11. Common Refinement
  12. Expanding Front
  13. Integration over Curved Polygons
  14. Numerical Experiments
  15. Conclusion
  16. ...and 6 more sections

Key Result

Theorem B.1

Consider three smooth curves $b_0, b_1, b_2$ that form a closed loop: $b_0(1) = b_1(0)$, $b_1(1) = b_2(0)$ and $b_2(1) = b_0(0)$. Take any smooth map $\varphi(s, t)$ on $\mathcal{U}$ that sends the edges to the three curves: Then we must have for antiderivatives that satisfy $H_x = V_y = F$. When $\det(D\varphi) > 0$, this is just the change of variables formula combined with Green's theorem.

Figures (21)

  • Figure 2.1: Cubic Bézier triangle
  • Figure 2.2: The Bézier triangle given by $b(s, t) = \left[ (1 - s - t)^2 + s^2 \; \; s^2 + t^2 \right]^T$ produces an inverted element. It traces the same region twice, once with a positive Jacobian (the middle column) and once with a negative Jacobian (the right column).
  • Figure 2.3: Intersection of Bézier triangles form a curved polygon.
  • Figure 3.1: Mesh pair: donor mesh $\mathcal{M}_D$ and target mesh $\mathcal{M}_T$.
  • Figure 3.2: All donor elements that cover a target element
  • ...and 16 more figures

Theorems & Definitions (2)

  • Theorem B.1
  • proof