Robust radial basis function interpolation based on geodesic distance for the numerical coupling of multiphysics problems

TL;DR

This work proposes a modification to rescaled localized radial basis function (RL-RBF) interpolation to account for the geometry of the interpolation domain, by yielding conformity and fidelity to geometrical and topological features.

Abstract

Multiphysics simulations frequently require transferring solution fields between subproblems with non-matching spatial discretizations, typically using interpolation techniques. Standard methods are usually based on measuring the closeness between points by means of the Euclidean distance, which does not account for curvature, cuts, cavities or other non-trivial geometrical or topological features of the domain. This may lead to spurious oscillations in the interpolant in proximity to these features. To overcome this issue, we propose a modification to rescaled localized radial basis function (RL-RBF) interpolation to account for the geometry of the interpolation domain, by yielding conformity and fidelity to geometrical and topological features. The proposed method, referred to as RL-RBF-G, relies on measuring the geodesic distance between data points. RL-RBF-G removes spurious oscillations appearing in the RL-RBF interpolant, resulting in increased accuracy in domains with complex geometries. We demonstrate the effectiveness of RL-RBF-G interpolation through a convergence study in an idealized setting. Furthermore, we discuss the algorithmic aspects and the implementation of RL-RBF-G interpolation in a distributed-memory parallel framework, and present the results of a strong scalability test yielding nearly ideal results. Finally, we show the effectiveness of RL-RBF-G interpolation in multiphysics simulations by considering an application to a whole-heart cardiac electromecanics model.

Figures (9)

• Figure 1: (a) Examples of domains with complex geometrical features. The red dots mark points on the domains that are close in the Euclidean metric, but that belong to topologically distant regions. The heart domain on the right is the Zygote heart model zygote. (b) A schematic representation of the discrete geodesic distance. (c) An example where, even if $\mathbf x \neq \mathbf y$, $g_h(\mathbf x, \mathbf y) = 0$.
• Figure 2: Idealized benchmark (\ref{['sec:ring']}). (a) Interpolation domain $\Omega$, overlaid with the meshes $\mathcal{M}^\text{src} = \mathcal{M}^\text{tet}_{\text{ring,1}}$ (red) and $\mathcal{M}^\text{dst} = \mathcal{M}^\text{hex}_{\text{ring,1}}$ (black). (b) Source data $f(x, y, z) = \mathop{\mathrm{arctan_2}}\nolimits(z, -x)$. (c) Interpolant obtained without geodesic thresholding. Notice the different colorbar scale and the spurious oscillations close to the slit. (d) Interpolant obtained from mesh $\mathcal{M}^\text{tet}_{\text{ring,1}}$ with geodesic thresholding.
• Figure 3: Idealized benchmark, convergence test (\ref{['sec:convergence']}). Interpolation error, defined as in \ref{['eq:error']}, against the maximum element diameter $h^\text{src}_{\max}$ of the source mesh $\mathcal{M}^\text{src}$. The interpolant was computed setting $M = 1$ (left) and $M = 4$ (right), for different values of $\alpha$. Different choices of $\alpha$ correspond to different colors, while circles and crosses correspond to interpolants without and with geodesic thresholding, respectively.
• Figure 4: Idealized benchmark, scalability test (\ref{['sec:scalability']}). Total wall-time (left) and parallel speedup for different steps of the construction and evaluation of the RL-RBF-G interpolant.
• Figure 5: Schematic representation of the electromechanics model of \ref{['sec:heart']}, reporting the multiphysics interactions.

Theorems & Definitions (4)

• Definition 1: Geodesic distance
• Definition 2: Discrete geodesic distance
• Definition 3: Discrete geodesic distance
• Definition 4: Thresholded Euclidean distance